Ben-Gurion University of the Negev Faculty of Engineering Sciences Department of Industrial Engineering & Management

BEN-GURION UNIVERSITY OF THE NEGEV FACULTY OF ENGINEERING SCIENCES DEPARTMENT OF INDUSTRIAL ENGINEERING & MANAGEMENT

Momentum in Football

THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE M .Sc DEGREE THE M.Sc DEGREE

By: Emri Dolev By: Emri Dolev

March, 2020 March, 2020

01.03.2020

_ 1/3/2020 i

Abstract:

Many studies have been conducted on momentum in sports, with contrasting results and their underlying explanations debated greatly. We define momentum to be a state where success in a previous event leads to an increased likelihood of success in the next corresponding event. The initial argument regarded the validity of calling momentum by its name, with the “Hot-Hand” fallacy (Gilovich, Vallone, & Tversky, 1985) stating that the momentum that is seen and perceived is nothing but a human heuristic which allows us to see patterns, even when those patterns are random. Curiously, unlike other sporting fields, this topic has hardly been studied in the world’s most popular sport – football. As part of this study we carried out an observational, time-series analysis of both team and player levels from the elite European football leagues. We discovered evidence for the existence of momentum carried over from match to match for football forwards as a result of providing goals in their previous appearance. On a team level, only a weak momentum effect was found – which could be better attributed to the teams’ qualities. We discuss the implications of these findings and propose future directions of research that should be conducted in order to further the research of momentum in football.

Keywords: Momentum, Football, Sports, Hot Hand, Mixed Logistic Regression,

Statistics

Acknowledgements

I would like to express my deepest gratitude to my two academic advisors – Professor Yisrael

Parmet and Professor Miki Bar-Eli for helping to guide me through the process of writing this thesis.

From our very first meeting together, you have been nothing but supportive and have been equally inspirational and motivational for me. I have thoroughly enjoyed our shared meetings, and the different topics and discussions that have arisen. At times these discussions could sway off-topic in relation to the subject-matter of this thesis, but none-the-less they were truly stimulating and fascinating. I could not have completed the writing of this thesis without your directions, whether Miki on the theoretical side or Yisrael on the methodological side. Thank you both so much for being excellent advisors to me.

I would also like to thank my family, especially my parents. I am so lucky and grateful to have such a supportive foundation that I can always count on and lean on to raise my spirits during hardships, but also to bring me down to earth when a little perspective is required. You have given me so much, even when I may not have deserved it or not known to appreciate it appropriately. You have provided me a full and plentiful life thanks to your hard work and love. I cannot put into words how much you both mean to me, and I do not say it enough – I love you .

Thank you for everything.

iii Table of Contents

Abstract (English) i Acknowledgements ii 1 Introduction 1 1.1 Motivation ...... 1 1.2 Theoretical Background ...... 3 1.2.1 Momentum in Sport ...... 4 1.2.2 “Hot Hand” Phenomenon ...... 8 1.2.3 Self-Efficacy ...... 18 1.2.4 Self-Efficacy, Psychological Momentum, Performance and Success 23 1.2.5 Psychological Momentum and “Hot Hand” Research in Football . . 31 1.3 Research Outline ...... 35 1.3.1 Research Questions ...... 36 1.3.2 Research Scope ...... 36 1.3.3 Project Outline ...... 37 2 Method 38 2.1 Data Sources ...... 39 2.2 Data Description ...... 39 2.2.1 Player-Level ...... 39 2.2.2 Team-Level ...... 42 2.3 Data Engineering ...... 44 2.3.1 Player-Level Data Engineering ...... 44 2.3.2 Team-Level Data Engineering ...... 54 2.4 Prepared Data ...... 72 2.5 Statistical Model ...... 73 2.5.1 Model Definition ...... 75 2.5.2 Model Selection ...... 76 3 Results 78 3.1 Player-Level Analysis ...... 78 3.1.1 Scoring a Goal is Considered a Success ...... 78 3.1.2 Scoring or Assisting a Goal is Considered a Success ...... 83 3.1.3 Mediation of Relationship by Minutes Played ...... 88

iv 3.2 Team-Level Analysis ...... 91 3.2.1 Winning Equals Success – with Teams’ Quality Gap ...... 91 3.2.2 Winning Equals Success – without Teams’ Quality Gap ...... 93 3.2.3 Comparison of the Team-Level Models ...... 96 4 Discussion 98 4.1 Player-Level ...... 99 4.1.1 Mediation by Minutes Played ...... 100 4.2 Team-Level ...... 101 4.3 Future Research ...... 103 4.4 Practical Application ...... 105 4.4.1 Comparison of Forwards Across Different Eras ...... 105 4.4.2 Scouting ...... 106 4.4.3 Managers’ Press Conference ...... 106 4.4.4 Man Management ...... 106 4.4.5 Story-Telling of Sports ...... 107 Appendices 108 The 95th Percentile of Scorers in the “Top-5” European Football A 108 Leagues...... B Historical Forward Legends of Football ...... 110 Teams that Played in “Top-5” European Football Leagues from C 111 2000-2019 ...... References 114 Abstract (Hebrew) 120

1 Chapter 1

Introduction

1.1 Motivation

There is a common notion of momentum in the world of sports, in which both players and teams can reach peaks of form that are driven by an enhanced psychological state as a result of their previous successful performances. This perception of momentum that drives future success is at the heart of this study as we set out to investigate whether it is grounded in empirical evidence, or simply an illusion conceived by the human imagination that has been imprinted into our narratives when discussing sports.

This belief in momentum is held at both a player-level, where a single athlete has their own momentum based on their personal performances and experiences, and also at a team-level where collective momentum is developed based on a team’s collective performances and experiences.

A team’s recent form is a central part of the conversation whenever commentators are previewing an upcoming match (“this team is coming into this match on the back of good/poor form”), or reviewing the outcome of a completed match (“this team is continuing its rich/dreadful vein of form”). Teams’ form is also a central statistic in the betting industry, with a visualization of a team’s recent results often being displayed to potential gamblers as part of the information provided to them regarding upcoming matches.

Yet, the most famous example of this observed momentum in sports is the “hot hand” phenomenon – a basketball player taking a shot is considered to be more likely to score if their recent previous shots were successful, and they are therefore considered to be in a

2 “hot streak” of scoring. Generally speaking, ”the belief is that the performance of a player during a particular successful period is significantly better than expected on the basis of the players’ overall record” (Gilovich, Vallone, & Tversky, 1985).

This belief also holds true in the world’s most popular sport, football, and is shared by commentators, spectators , players, and coaches/managers alike. The belief of momentum at a player-level is exemplified by former Manchester United manager Sir

Alex Ferguson’s explanation of an apparent “dip” in the form of Wayne Rooney, England’s all-time top international goalscorer in 2010. “I say it time and time again; strikers live by their goals. When they are not scoring, they think the goals are never going to come.

Then, when they do come, they think they are never going to finish. That is normal with any forward.” (The Telegraph, 2010).

Forwards in football are unique because they are tasked with the responsibility of providing the goals for their teams which are crucial in leading their teams to win.

Extended periods of matches where forwards play without scoring have been termed

“goal-droughts” by the media – an unproductive period that practically every top forward in the game has endured during their career, and is often a central talking point during managers’ press conferences.

Examples of top football managers’ belief in momentum when asked about their forward’s goal-droughts are plentiful. When asked about Romelu Lukaku’s goal drought in

2018, Jose Mourinho, one of the most successful managers in recent years replied, “Not just with the goals he is not scoring but also in his confidence, in his movement, his touch,” Mourinho told reporters after a defeat, “He is not linking the game well with the team. But he is our striker and he is a good striker we believe in. Romelu is a hard- working guy and a good professional. One day the goals will arrive and the confidence will be back.” (Reuters, 2018)

3 Sir Alex Ferguson, widely considered to be one of the greatest football managers of all- time due to his success and longevity, was the subject of a case study of his methods

(Elberse, 2013). Eight leadership lessons that captured the central elements of Ferguson’s managerial approach were outlined, and two of these leadership lessons are of major relevance to this research; (a) to match a message to the moment, (b) to rely on the power of observation.

As part of this study, our intention is to investigate whether momentum exists in football.

We will rely on statistical modelling of historical observations to investigate the idea that teams and/or forwards are affected by their recent “form”. Our aim is to discover if this perception of momentum is grounded in statistical significance or not – is there empirical evidence for its existence in football; perhaps managers simply attempt to instil confidence in their forwards during these goal-droughts by matching a message for them during their moments of poor “form”, perhaps teams do not develop collective momentum and the pundits, coaches, and supporters alike all subconsciously see artificial patterns in teams’ recent results which they mistakenly attribute to “form”.

1.2 Theoretical Background

The question whether momentum exists in sports remains both highly debated and unresolved, as are the possible explanations of the causes at the heart of each side of the argument. Dismissing the phenomenon of momentum not only challenges the perceptions of those directly involved in sports, but it also negates a whole body of research and theories regarding the effects of psychological sensations on performance enhancement. The upcoming section will outline the historical roots and research that have formed the foundations for the debate of the existence of momentum. We will present both the supportive and opposing testimonials for the existence of momentum in

4 sports, in addition to presenting the theories that outline the causes and theoretical explanation for each of the respective claims.

1.2.1 Momentum in Sport

The original research of momentum in sports was carried in the late 1970’s, attempting to uncover its role. The researchers observed different sports teams, and reviewed verbal and written accounts from athletes (Adler & Adler, 1978). Using this information, they constructed an explanation of the phenomenon of momentum in sport – focusing on momentum as an emotional aspect of sport, the researchers examined how and what occurs.

They defined positive momentum as a period of time marked by a feeling of invigorated confidence and determination; “the individual feels possessed with an ability for accomplishment beyond his normal equilibrium state” (Adler & Adler, 1978). Positive momentum was associated with periods of competition, such as a winning streak, when everything seems to ‘go right’ for the competitor. In contrast, negative momentum was associated with periods, such as losing streaks, when everything seems to ‘go wrong’.

1.2.1.1 Momentum Before and During a Game

“The structure of sport and games is such that preconditions can create situations where momentum exists prior to the start of a contest” (Adler & Adler, 1978). The main concept for the inception of momentum before a contest is “psych”; competitors may have fears, expectations, and hopes regarding how they see things unfolding during a contest.

Contestants will often try to “psych-up” themselves – to mentally attain a higher level of psychological energy in order to perform better, or to “psych-out” their opponents – trying to decrease their levels of psychological energy through fear and thus hopefully restraining their performances. Examples of methods competitors undertake to “psych-

5 themselves-up” included inducing hate for their opponents and carrying out pre-game rituals such as wearing special clothing or carrying out specific repeated routines.

Pre-game, an important psychological component is pressure, which commonly leads to fear. Pressure can be used by an athlete to “psych-up” as it gets adrenaline rushing and can induce outstanding performances, yet pressure can also become too much to handle in which case it could then lead to “psyching-out”. Pressure is therefore a key tool prior to a contest as one competitor will try to instil fear in their opponent in the hope that this will deteriorate their performance. Nowadays, in the age of globalization, the combination of wider digital broadcasting and the publishing of spectators’ opinions and remarks through social-media could possibly intensify pre-match pressure.

The principal factor of early in-game momentum is the environment; the weather, the location – is it inside or outside, the surface – grass or artificial-turf. Being familiar with the surroundings could lead to major advantages, as is often captured with home-court- advantage; playing at your home ground can very feasibly offer an advantage to the hosts, who are familiar with the facilities, have less travel time to the location, and are cheered-on by the majority of the supporters at the venue. This, along with possible

‘tension in the air’ – a feeling of added pressure due to the importance of a specific contest, can be telling in the swinging of momentum throughout a contest.

1.2.1.2 Momentum Starters, Maintenance, and Breakers

“Momentum can penetrate a contest either as a prearranged fact, through a gradual process, or as a flash of light” (Adler & Adler, 1978). Charismatic plays like a dunk in basketball or specific actions by a charismatic player like a crunching slide tackle in football, can often be catalysts for momentum, changing the rhythm of a contest, and stimulating one side. “Taking a chance” – an action that is against the odds or daringly

6 unexpected can also provoke momentum, whether the action is successful or not. For example, a player taking a shot from outside of the box in football is unlikely to result in a goal, yet it often boosts the morale of the onlooking supporters and the team who managed the shot on goal. On the other hand, missing a big chance of scoring or winning often directs momentum in the direction of the opposition. Creating a scene during a contest can enflame the atmosphere of a contest and also create momentum – for example the break-out of a fight, coaches and players berating the officials, or physical encouragement of the crowd to be more vocal in their support.

Territoriality is a significant aspect of team sports and dominating it can enhance momentum for the team who is playing in their attacking territory, often at the expens of the opposition’s momentum. Other momentum triggers include the collective call of the audience, luck – a lucky escape from conceding or leniency from a referee, and finally

“time panic” – when momentum sways as the contest enters its final stages.

“In most cases momentum swiftly subsides, but without having been maintained for some actual length it cannot be considered genuine” (Adler & Adler, 1978). The ability to sustain the momentum that has been attained is of major importance, it is no good having momentum and losing it before it is materialised into a tangible advantage in the score-line. In moments of positive momentum there are elements of rhythm, pace, and tempo; players feel like they can’t miss, play is fluid and effortless, confidence is high, and the contestants glide gracefully over the field. Yet when the momentum is low teammates shy away from receiving the ball, taking responsibility and shooting; play is rough and choppy, possession is awkward and frequently broken. “The ability to capitalize on other players’ mistakes is critical, by following up an opponent’s errors with successful manoeuvres one can generate added velocity and intensity to momentum” (Adler &

Adler, 1978). For example, in football, intercepting a pass high up the field as the

7 opposition is trying to build an attack leaves them short of numbers in defence and vulnerable to concede a goal. If capitalized on, it will leave the opposition demoralized and could possibly cause friction as blame is cast on the player who lost the ball originally.

Exploiting the momentum while you have it is often the responsibility of the coach; “the coach dictates the tone of play, knowing when and where to direct his athletes’ abilities to their maximum, winning the desired momentum and best results” (Adler & Adler,

1978).

The two main momentum breakers are interruption and intrusion; this can be any event that disrupts the tempo. During games this can come in the form of calling a time-out, time-wasting, or the simulation of fouls and subsequently staying down for an extended time. Undesirable incidences can also break momentum; embarrassment or humiliation may fluster teams into losing their cool. The loss of a key player through injury will obviously leave a void of talent but could also demoralize the teammates who are left to battle without one of their most important figures.

1.2.1.3 Seasonal Momentum

John Wooden, a renowned basketball coach is quoted as saying, “For every artificial peak you create there is a valley, and I don’t like valleys. Games can be lost in valleys. The ideal is an ever-mounting graph that peaks with your final performance”. The best players will use momentum to conquer these peaks and try to minimize the effects of the valleys.

Momentum obviously varies throughout a season as an individual or team may streak or slump. Athletes must prepare themselves for this; yet self-control and emotional-stability is both vital and hard to master. “While ‘hot’, a player feels great – confident and effervescent. Conversely, when things go poorly, anxiety and depression overshadow the home and locker room atmosphere” (Adler & Adler, 1978).

8 No winning streak lasts forever, the loss that breaks the streak is only a matter of time, and the subsequent emotional let-down unavoidable. Therefore, resiliency – the ability to bounce back, is imperative. Momentum breakers throughout the season are plentiful and can come from many sources; being rested for a match, prolonged injury, personal issues at home, disagreements with the management due to either tactics or contract disputes, weather conditions that affect style of plays, team ‘chemistry’ or ‘spirit’ being disturbed by locker room quarrels, and trades or transfers.

“In extremely rare instances, a team can possess such a strong binding force as to exhibit multiyear momentum. It is here that the ultimate of team play can be found, when a tradition of excellence is established: the dynasty” (Adler & Adler, 1978). With regards to this study, in footballing terms it is hard to look past Sir Alex Ferguson’s 27-year, trophy- laden tenure at Manchester United as the prime example of a footballing dynasty.

Confidence is a prime element of momentum (Adler & Adler, 1978), and exists in two forms; (i) short-term anticipated success based on immediate occurrences during a contest, (ii) a long-term anticipation of success based on previous success or a strong belief in one’s inherent ability. Maintaining momentum then becomes dependent on the athletes’ level of perseverance and concentration. Yet, the most important factor for possessing momentum remains the initial catalyst for its emergence – the occurrence that creates it in the first place. Without it, momentum will never emerge.

1.2.2 The “Hot Hand” Phenomenon

The “hot-hand” phenomenon depicts the common belief that is recognized in sports as

‘streakiness’; an athlete’s probability of success is greater following a previous success rather than following a failure. Debate surrounding the statistical validity of this phenomenon has been contended for decades and is still ongoing today. The rebuttal of

9 the “hot-hand” phenomenon rejects the role of previous success and the effects of self- efficacy in the enhancement of future athletic performance (Bar-Eli, Avugos, & Raab,

2006). This debate has been raging for well over 30 years now, with constant new research attempting to come to a conclusive answer.

1.2.2.1 The “Hot-Hand” Fallacy

The original study that led the challenge to the “hot-hand” phenomenon, its theoretical foundations, and ignite the on-going debate around momentum was published in the mid

1980’s. It sought out to investigate the origin of the extensive belief in the phenomenon, and its statistical validity.

As the name of the phenomenon suggests, basketball was chosen as the sporting area of research – specifically, the “hot-hand” describes the belief that a “basketball player’s chance of hitting a shot are greater following a hit than following a miss on the previous shot” (Gilovich, Vallone, & Tversky, 1985). This opinion expresses the idea that the performance of a player during a successful period is significantly better than his normal, base-line performance.

The researchers carried out three different studies of the phenomenon, yet found no statistical validity for the belief. They branded the “hot-hand” a fallacy; stating that the detection of perceived ‘streakiness’ was just an example of the representativeness heuristic and attributed the misconception to a general inability of the population to comprehend randomness. The representativeness heuristic is defined as "the degree to which an event (i) is similar in essential characteristics to its parent population, and (ii) reflects the salient features of the process by which it is generated" (Kahneman &

Tversky, 1972). The heuristic leads people to attribute erroneous, subjective probabilities to future events with the purpose of making judgmental shortcuts under uncertainty.

10 The simplest example of representativeness can be described by a sequence of flipping a fair coin; 50% probability of heads, 50% probability of tails. People expect short sequences of heads and tails outcomes to reflect the fairness of the coin, and therefore a

‘streak’ of heads increases the probability of tails in the next flip, as the fairness regresses the results back to the mean – 50% probability of each outcome. This conception holds true when considering the law of big numbers – approaching infinite coin flips, yet the misconception is that this law also holds true for small numbers i.e. short sequences of coin flips. This representativeness produces two related biases; (a) the gamblers’ fallacy

(Tversky & Kahneman, 1974) – the belief that the probability of heads is greater after a sequence of tails than after a sequence of heads, (b) it leads individuals to reject the randomness that exists in sequences (Gilovich, Vallone, & Tversky, 1985).

Basketball offered an interesting context for researching the perceptions of randomness.

For example, a basketball player may be considered ‘streaky’ by spectators when he hits four or more shots in a row – yet such sequences can only be attributed to an enhanced psychological state if the length or frequency of such sequences exceeds what would be expected statistically. The implications of this in basketball were two-fold; “(i) the probability of hit should be greater following a hit than following a miss, (ii) the number of streaks of successive hits or misses should exceed the number produced by a random process with a constant hit rate” (Gilovich, Vallone, & Tversky, 1985).

The first study in the original “hot-hand” research set out to investigate how broad the belief in the “hot-hand” was for basketball fans. A sample of one hundred basketball fans was assembled, all of them playing basketball occasionally and watching at least five games a year. The fans were surveyed regarding the sequential dependence among shots, and revealed that basketball fans believe in “streak shooting”: 91% of them believed that that a player has “a better chance of making a shot after having just made his last two or

11 three shots than he does after having just missed his last two or three shots”; 68% of the fans expressed essentially the same belief for free throws, claiming that a player has “a better chance of making his second shot after making his first shot than after missing his first shot” (Gilovich, Vallone, & Tversky, 1985). Similar beliefs were also expressed by professional players.

After showing that the ‘hot-hand’ belief was prevalent in fans, the researchers set out to see if these beliefs were based on statistically significant phenomena. The Philadelphia

76er’s shooting data was collected for nine players from 48 matches in the 1980-1981 season. An analysis of conditional probabilities was carried out – do players have higher percentages of successful shots given they just made a successful shot compared to when they have just made missed their last shot. Their results found that the only trend in the data actually contradicted the predictions of the “hot-hand” hypothesis.

An analysis of runs statistical test was then carried out – a run being a sequence of consecutive successes or failures. The number of runs observed was then compared statistically to then number expected if all shots were assumed to be independent of each other. They found that only one of the nine players showed significant difference between the number of runs observed and expected. Their final test applied to the 76er’s data tested the stationarity of the data; does the probability of shooting success vary during matches when the player is “hot”, and his performance elevated temporarily. The results provided no statistical evidence for “streakiness” and non-stationarity.

Similar tests were carried out on free-throw data from the Boston Celtic’s players during two seasons. Free-throws are unopposed attempts, typically taken in pairs, to score by shooting from the same location behind the free throw line and are generally awarded after a foul on the shooter by the opposing team. The researchers claimed that free- throws would provide a better test for independence between successive shots as they

12 offer better replication validity – free from confounding effects of opposition defences and shootting locations. Once more, the results were the same; they found “no evidence that the outcome of the second free-throw is influenced by the outcome of the first”

(Gilovich, Vallone, & Tversky, 1985).

1.2.2.2 Subsequent “Hot-Hand” Research in Sports

The debate over the effects of previous success on the future has ensued ever since the original “hot-hand” study provocatively claimed it was a cognitive illusion resulting from simple human heuristics. Gilovich et al.’s (1985) research has generated further analyses that have been carried out for over 30 years, in many different sporting fields, with the inconclusive results. Some studies show evidence for the phenomenon and others against, with some dependency on the type of sport – individual or team. Academic reviews of the state of the art “hot-hand” research have found that empirical evidence for its existence was considerably limited (Bar-Eli, Avugos, & Raab, 2006) , and largely supported the claim that the “hot-hand” does not exist in sports (Avugos, Köppen,

Czienskowski, Raab, Bar-Eli, 2012).

1.2.2.2.1 Evidence Against the Existence of “Hot-Hand”

Following on from the declaration by Gilovich et al. that the “hot-hand” phenomenon was a fallacy, many future studies, stimulated by its conclusions, have shown similar results.

An analysis of four years’ worth of baseball data, from 1984 to 1987, compared the batting averages of players in matches following “hot” and “cold” streaks – a sequence of five games over less than seven days, during which the player had a batting average of more than 0.400 or less than 0.125 hits per bats, respectively. The results found no significant differences between the batting averages which were just as likely to be higher following “cold” streaks as following “hot” streaks (Siwoff, Hirdt, & Hirdt, 1988). Another

13 baseball study examined 501 players’ batting records over four different seasons from

1987 to 1990, found that some batters did exhibit streakiness in some of the seasons, but did not do so consistently, and the number of runs (sequences of successful or unsuccessful at-bats) did not significantly deviate from what was expected from randomness (Albright, 1993). Similar results were reported for sequences of consecutive hits at team level in baseball (Frohlich, 1994). A review of Albright’s (1993) study, argued that the evidence they found for individual streakiness was not statistically convincing and proposed analysing the data using another statistical model. Yet the different technique resulted in the same results and concluded that streakiness does not exist in batting (Stern & Morris, 1993).

Further studies in the search for the “hot-hand” in basketball have also debunked the phenomenon. A study of 19 National Basketball Association (NBA) games considered the amount of time between shots during sequences, hoping to find the elusive “hot-hand”.

Yet the results once more negated the perceptions of the phenomenon as the mean time interval between two successful shots was longer than the mean time between an attempted shot and following a missed shot (Adams, 1992). Another study suggested that the NBA Long Distance Shootout contest, an annual event in which eight of the best 3- point shooters in the NBA compete against one another, provided better context than free-throws for uncovering the “hot-hand” in basketball. Yet, once again no significant streaks of success nor any sequential dependence were detected after comparing expected and observed runs for 21 of the 23 players analysed over four annual contests from 1994 to 1997 (Koehler & Conley, 2003).

A combined study of baseball and basketball was carried out in 2000, extending the search for the phenomenon by looking for momentum over the length of a season. Data was collected from 28 Major League Baseball (MLB) and 29 NBA teams. The researchers

14 compared the observed winning and losing streaks to what would be expected assuming that each game’s outcome was independent of the result of the previous game. The statistical tests showed a close fit between the observed and expected streaks under the independence assumption, once again contrary to the “hot-hand” belief; the probability of winning or losing a game does not depend on the team’s previous performances

(Vergin, 2000).

The search for the phenomenon has also been carried out in golf, where data from the

Professional Golfers' Association (PGA), In 1997, data was collected for 35 golfers recording their hole-to-hole scores. The scores were converted to below or above par and arranged in a 2x2 contingency table. The researchers looked for evidence both within individual tournaments and between them during the entire season. Yet, no statistical evidence for the “hot-hand” was found and strongly suggested that past performances is not a good predictor of future success (Clark, 2005).

A meta-analysis of “hot-hand” was carried out in 2012 and included 30 academic studies that provided an empirical investigation of the phenomenon in sports. The analysis of the effects found a mean effect size very close to zero, leading the researchers to argue against the existence of “hot-hand” (Avugos et al., 2012) and provided additional support for Gilovich et al.’s (1985) claim.

1.2.2.2.2 Evidence in Favour of the Existence of “Hot-Hand”

The findings of the original Gilovich et al. research have also been challenged, with subsequent studies continuing the search for the “hot-hand” and showing that the perceptions of it are indeed justified. “It is worth noting that the strongest support for the

“hot-hand” phenomenon was found in the more ‘individually’ performed sports (as opposed to the team-sports)” (Bar-Eli, Avugos, & Raab, 2006).

15 An analysis of shots attempted, and made, by all NBA players during the 1989-1990 season included data collected for 123 players who had played in over fifty games and averaged more than nine shots per game. The idea of the study was to categorise the streak shooters into three categories; those with (i) “hot and cold streaks”, “cold streaks”,

“hot streaks”. Streak shooters were defined as players whose performances in at least 5% of the matches they were involved in were inconsistent with their overall season shot success percentage. Results showed that 17 players exhibited streakiness, especially three

(Forthofer, 1991).

Several years later, a study revisited Forthofer’s (1991) idea and, by eliminating the reconsidered weaker cases, found only 10 streaky players, and suggested that the analysis be carried out over several years to test whether the streakiness was consistent in time

(Wardrop, 1998). The same Wardrop had earlier re-examined Gilovich et al.’s (1985) free- throw data and showed that if each player’s data is aggregated then the fan’s perceptions of the “hot-hand” are true; Shooting success increased from 74% after a missed shot to

79% after a successful one (Wardrop, 1995). This aggregation though was not stated to be proof of the “hot-hand” phenomena but rather an insight into why it is prevalent, asserting that it is an example of Simpson’s paradox (Simpson, 1951) – which shows that people detect patterns, even where none exist.

The emergence of larger datasets has enabled more powerful statistical analyses to be implemented. An analysis of five years’ worth of NBA data and 300,000 shooting attempts found that the second free-throw attempt’s success rate is significantly higher following by a successful first attempt in comparison to a missed first attempt. The conclusions were drawn at both aggregated levels, and individual (Yaari & Eisenmann, 2011).

Evidence for the “hot-hand” was shown in golf-putting and dart-throwing (Gilden &

Wilson 1995). Results showed that the magnitude of a streak is closely related to task

16 difficulty – measured by the success rate. The search for streakiness in golf continued, and analysis of professional golfers from the PGA Tour and the Senior PGA Tour over two years carried out. Streakiness was found but was explained by the difficulty of the golf courses rather than the because the golfers themselves were affected by any momentum

(Clark, 2003a).

Individual sports seem like better contexts to explore in the quest to find the “hot-hand”, in these sporting environments an athlete is only dependent on his own performance in direct comparison to that of their opponent; psychological strength and character is likely to be more crucial to attaining an advantage.

In 2001 data from 90,000 points during 481 Wimbledon matches from 1992 to 1995 was analysed in search of the “hot-hand’ in tennis. Significant statistical evidence was shown to support the claim that winning the previous point influenced the probability of winning the next point, though the effect was weak (Klaassen & Magnus, 2001). Furthermore, the strength of the effect was shown to be negatively correlated with the player’s strength for both men and women.

Different individual sporting fields have been explored for the phenomenon. It is suggested that it is more likely to emerge in sports where the performances are more uniform (Oskarsson, Van Boven, McClelland, & Hastie, 2009) and almost identical – in these more repetitive sports the athlete has more command of outcomes and the confounding factors are better controlled. Examples include darts, horseshoe pitching, bowling, and billiards. Horseshoe pitching was a prime sport; every pitch is from the same distance at regular intervals, with intense concentration and no strategy. Data was collected from the 2000 and 2001 World Championships. The analysis indicated that both elite male and elite female pitchers do indeed have “hot” and “cold” spells. The variations found in the pitchers’ performances both within games and within tournaments provided

17 evidence for the phenomenon in horseshoe throwing as success in the previous throw affected the probability of success in the next (Smith, 2003). Bowling was also researched, again ideal as it’s another individual, repetitive sport. Data from the Final Rounds of the

Professional Bowlers Association (PBA) tournaments was collected from 1994-1998.

During these years the final round was played as a knock-out format where one-on-one matches were played and the winner of the match went onto the next round where he played a higher-seeded opponent. Results indicated that winning the previous round increased the probability of winning the next by more than 50%, even though the opponents were seeded higher (Frame, Hughson, & Leach, 2003). Professional bowling was revisited again, with the analysis of season data from 2002-2003 and searching for streakiness. Results showed that the probability of rolling a strike (knocking down all 10 pins in the first of the two rolls in a turn) was dependent on the outcomes of the previous turn, and the number of strikes rolled in matches exceeded what was expected purely by randomness (Dorsey-Palmateer & Smith, 2004).

A final consideration of the phenomenon takes an epistemologist’s viewpoint of the hot hand in sports who “argued that being hot does not have to do with the fecundity, duration, or even frequency of streaks. It has to do with their existence. The conclusions to be drawn are (a) one has a hot hand when one is playing better than average; (b) players often know when they are playing better than average; and (c) observers can often tell when players are playing better than average. This judgment of countless fans, coaches, and players is vindicated.” (Hales, 1999).

1.2.2.2.3 Statistical Issues in “Hot-Hand” Research

Choosing the correct statistical analysis to apply in attempting to uncover “hot-hand” is obviously crucial. Arkes (2013) found that prior studies claiming that the “hot-hand” phenomenon does not exist were not likely to be able to discover “hot-hand” because of

18 two reasons: (i) small sample sizes and a consequent lack of statistical power and (ii) failure to consider the frequency of the “hot-hand”. Momentum is not maintained for long periods and is an infrequent event that is difficult to detect. Arkes (2013) concluded by stating that results from larger datasets showed that the “hot-hand” effect is strong but infrequent. The likelihood of detecting the “hot hand” increases as it becomes more frequent, yet this likelihood will often be small if the frequency of “hot hand” is small, no matter how large the dataset is.

Evidence for “hot-hand” is virtually prevented from being observed, especially for studies with small samples (Stone, 2012). Stone showed that measurement errors occur in basketball; “shots made” measures probabilities with error, and the autocorrelation between shots is grossly underestimated. Stone (2012) showed that if the measurement error bias is removed from previous basketball studies that found evidence against the phenomenon, then strong evidence for the existence of “hot-hand”.

1.2.3 Self-Efficacy

The foundations of this debate over momentum were laid in the late 1970’s with the introduction of the concept of self-efficacy as part of a unifying theory of behavioural change to overcome adverse situations or tasks. Self-Efficacy was defined as “a person’s conviction that they can successfully execute the behaviour required to produce the outcomes” (Bandura, 1977). In simpler terms, self-efficacy is a measure of how much belief a person has in their own ability to succeed in specific situations or accomplish a task – for example, “I am confident that I can score a goal in the match today”. Different levels of self-efficacy can determine how a person will approach and manage different challenges.

19 The theory stated that psychological procedures and experiences alter the level and strength of self-efficacy and investigated its sources. The results of the study, which opened the door for a wide range of research in the field, showed that “self-efficacy was a uniformly accurate predictor of successful performance on tasks varying in difficulty”

(Bandura, 1977).

1.2.3.1 Strength of Self-Efficacy

The strength of the belief that one can perform a task well will “affect both the initiation and persistence of coping behavior” (Bandura, 1977). For example; people with high self- efficacy who hold strong beliefs of personal mastery are more likely to view difficult tasks as challenges to overcome rather than something to be avoided – they will expend more effort in the task and sustain it for longer periods of time. The persistence and perseverance in activities that are subjectively intimidating to the person, but are in fact relatively safe, will produce a further increase in their self-efficacy through their experience of overcoming the task. Conversely, the opposite holds true for failures in a task, which will diminish one’s self-efficacy.

According to the theory “self-efficacy varies on several dimensions that have important performance implications” (Bandura, 1977). There are three dimensions of a person’s self-efficacy that make it more comprehensive and give a better indication of how much self-efficacy a person possesses; (a) Magnitude – how a person’s self-efficacy varies according to the difficulty of the task at hand. Therefore, it is natural that people have higher self-efficacy for simpler tasks compared to more difficult ones (b) Generality – a person’s self-efficacy can be very specific to a given task, or can be more generalized to different tasks, situations, and domains. (c) Strength – How resilient one’s self-efficacy is in the face of failures. Weak self-efficacy in a task is readily demoralized by failing to master it, whereas strong self-efficacy enhances perseverance in the face of failure.

20 1.2.3.2 Sources of Self-Efficacy

The theory stated that people acquire their self-efficacy from four major sources

(Bandura, 1977), with the varying levels of influence. The sources, in order of influence, were (i) performance accomplishments, (ii) vicarious experience, (iii) verbal persuasion, and (iv) emotional arousal.

i. Performance Accomplishments

This is the most robust source of self-efficacy and “is especially influential because it is based on personal mastery experiences”. Success in tasks increases self-efficacy while failure decreases it. Yet the timing and the pattern of the experiences plays a crucial role; failures lower self-efficacy more intensely if they occur in early attempts of a task, but if

“strong self-efficacy is established through repeated success, the negative impact of occasional failures is likely to be reduced”.

ii. Vicarious Experience

Observers who witness others perform tasks that are subjectively fearful for themselves, without any negative consequences, can generate beliefs that they too can carry out the tasks and improve through persistence.

iii. Verbal Persuasion

People can be encouraged and reassured to believe that they can cope with fearful situations successfully. Verbal persuasion is commonly used to influence the behaviours of others. This is due to the fact that it is easy to perform and can be carried out essentially by anyone who is seeking to persuade.

21 iv. Emotional Arousal

When judging one’s levels of anxiety and vulnerability to stress, people will often calculate their ability to overcome fearful situations as a function of the physiological arousal.

Of the four sources of self-efficacy, the most powerful is performance accomplishments.

The source is especially powerful when the accomplishment is achieved independently.

Independent achievement produces success experiences, which further reinforce expectations of self-competency. Furthermore, additional exposure to formerly aversive tasks lowers the emotional arousal the next time that the tasks are met – the independent mastery lowers stress as it provides additional opportunities to perfect coping skills. Verbal persuasion is a less powerful source of self-efficacy and the extent of its effect depends on the perceived credibility of the persuader. The higher their

“prestige, trustworthiness, expertise, and assuredness” the more likely they are to increase self-efficacy in the person they are trying to persuade.

1.2.3.3 Development of Self-Efficacy Over the Lifespan

The development of self-efficacy over a person’s lifespan were researched in the mid

1990’s by Bandura himself, as he continued to revisit the ever-growing and influential concept in psychology research. He presented a description of the natural dynamics of self-efficacy as one matures; from the beginnings as a new-born, to childhood, adolescence, adulthood, and finally advancing age (Bandura, 1994) – different coping skills and demands are required for each different period of life.

Babies are born without any sense of self, they explore their surroundings and observe the outcomes of their actions to acquire knowledge, this provides them an initial platform for developing their self-efficacy. As they accumulate personal and social experiences,

22 ”they will eventually form a symbolic representation of themselves as a distinct self”.

(Bandura, 1994)

As they move into increasingly larger social groups such as kindergartens, their peers gradually become more important in the development of their self-efficacy. The classmates provide informative comparisons for judging and verifying one’s self-efficacy, while the cognitively advanced children become role-models for effective behaviours and logical thinking. As they grow up and advance through the grades of primary school, the young students’ self-efficacy affects their aspirations, their levels of interest in different activities, and their accomplishments.

Entering their adolescence and approaching the demands of adulthood, they must now

“learn to assume full responsibility for themselves in almost every dimension of life. This requires mastering many new skills and the ways of adult society” (Bandura, 1994). The strength of their self-efficacy is enhanced by dealing successfully with potentially troublesome events in which they are inexperienced. The ease of the transition from childhood to of adulthood depends on the strength of the personal efficacy they have accrued through prior experiences.

As adults, people have to learn to cope with many new demands; independence, romantic relationships, careers, and possible parenthood. Once again, strong self-efficacy is a major factor in the ability to learn new skills, competencies, and being successful.

Those who enter adulthood with low self-efficacy and doubting themselves can often find the new demands stressful and depressing. Self-efficacy is a major factor in career development and success. In the early stages of one’s career people’s self-efficacy will contribute to the development of fundamental skills required for successful careers; interpersonal, cognitive, self-management and regulation. The development of these

23 skills will enhance coping capabilities and assist in managing one's motivation, emotional- states and thought-processes. This consequently increases self-efficacy furthermore.

“By the middle years, people settle into established routines that stabilize their sense of personal efficacy in the major areas of functioning” (Bandura, 1994). The stability of self- efficacy at work can be challenged by new external challenges; technological advancements such as computers, social transformations like the introduction of social- networks, and pressure coming from younger challengers looking to replace them at the workplace. “Many physical capacities do decrease as people grow older, thus, requiring reappraisals of self-efficacy for activities in which the biological functions have been significantly affected. However, gains in knowledge, skills, and expertise compensate some loss in physical reserve capacity.” (Bandura, 1994).

1.2.4 Self-Efficacy, Psychological Momentum, Performance and Success

1.2.4.1 The Relationship between Self-Efficacy and Psychological Momentum

Adlers’ (1978) research on momentum in sports clearly suggested that perceptions of momentum do exist psychologically for athletes and are volatile in response to different experiences. Yet, only a few quantitative studies have been carried out exploring the relationship between psychological momentum and self-efficacy, with encouraging results.

One study (Shaw, Dzewaltowski, & McElroy, 1992), examined self-efficacy and causal attributions as mediators of perceived psychological momentum. Participants were paired with an opponent and performed three sets of 10 basketball free throws. Random assignment was implemented to either a repeated success or a repeated failure group, manipulated by the quality of each participant’s opponent. Free throw self-efficacy, perceived psychological momentum, and causal dimensions were measured and assessed

24 after each set. As expected, their results showed that the experience of competitive success increased perceptions of psychological momentum while experiencing failure decreased psychological momentum. But “self-efficacy only changed in response to competitive success as the participants became more confident” (Shaw, Dzewaltowski, &

McElroy, 1992).

Another study (Mack & Stephens, 2000), tested the prediction that an event that changes psychological momentum will produce a corresponding change in self-efficacy. The study consisted of 101 university students shooting basketballs who were measured for psychological momentum and self-efficacy during the experiment, along with persistence, affect, and arousal. Their results showed that psychological momentum led to a shift in the same direction for self-efficacy and affect; subjects with positive psychological momentum had higher levels of self-efficacy and more positive thoughts, while the opposite was observed for negative psychological momentum.

1.2.4.2 Performance Implications of Negative Psychological Momentum

Fluctuations in psychological momentum can lead to reciprocal changes in athletes’ sense-of-control, their confidence, optimism, motivation, and energy (Crust & Nesti,

2006). Negative psychological momentum has the potential to take over a player’s psyche, as athletes psych themselves out by investing too much time and energy in attempting to overcome its effects. The negative momentum affects them both psychologically and physiologically; manipulating what they think, how they feel, and consequently altering their behavior. The loss of psychological momentum is often accompanied with a drop in their focus (Markman & Guenther, 2007). The athletes will often start replaying mistakes in their mind and over-analyzing their performances. A fear of not making further mistakes and an urge to make up for their errors can often lead them to intensify their efforts, sometimes too much. This combination of over-thinking

25 and trying too hard typically leads to poorer performances and has the opposite outcome to what was intended as the athlete makes more mistakes (Csikszentmihalyi, 1990).

An example for this in football can be found in a famous metaphor, believed to have been conceived by renowned Dutch forward Ruud van Nistelrooy, with the comparison of goals to ketchup. This metaphor has been repeated by renowned football figures like Jose

Mourinho, Gonzalo Higuain, and Cristiano Ronaldo, “Goals are like ketchup. Sometimes as much as you try, they don’t come out, and when they come, many come all at the same time.” (Laxmidas, 2010).

1.2.4.3 Self-Efficacy and Goal-Setting

In the early 2000’s, a team of researchers set out to explore how to influence, predict, and explain performance in work-related tasks. Based on over nearly four decades’ worth of empirical research, a theory of goal-setting was conceived – the theory of goal-setting and task-motivation (Locke & Latham, 2002). The theory was formulated based on the principle that conscious goals affect action (Ryan, 1970), with a goal defined as the aim of an action to attain a specific standard of outcome, usually within a specified time limit.

Their research showed that setting goals led to enhanced performance, in comparison to the customary message of urging employees to ‘do their best’. This is due to the fact that setting specific goals reduces uncertainty about what is to be obtained and sets a tangible standard of what is expected to be accomplished. Locke & Latham (2002) found that setting goals affects performance through four mechanisms; (i) goals give direction – they concentrate attention and effort towards goal-relevant activities, (ii) goals galvanize

– difficult goals extract greater effort and energy from employees, (iii) goals affect persistence – time constraints and deadlines affect work intensity, (iv) goals affect action indirectly – they activate cognition as employees engage in task-specific knowledge and

26 strategies. When given goals, employees will use relevant skills and knowledge for the task at hand. Yet, when the tasks given to the employees are new, employees will plan and establish strategies to overcome the challenge – with their level of self-efficacy determining how likely their strategies are to be effective and lead to success.

1.2.4.3.1 The Goal-Performance Relationship

The researchers discovered three moderators of the goal-performance relationship; (a) feedback, (b) complexity, and (c) commitment. Feedback is vital as it gives employees an evaluation of their progress – without it, it is difficult for them to adjust their behavior in order to increase the likelihood of accomplishing their goal. The employee’s self-efficacy is critical when receiving negative feedback and determines whether the performance appraisal is effective. As task complexity grows, more advanced skills are required, and higher level strategies need be conceived in order to achieve the goals. Because these abilities vary greatly between employees, the effects of goal-setting are greater for simple tasks and weaker for complex tasks. (c) The most important moderator of the goal- performance relationship is commitment; the relationship is strongest when people are committed to their goals and is especially important when the goals are difficult as they require increased effort.

Employees’ commitment is a factor of two categories; self-efficacy – their belief that they can attain the goal, and perceived importance of attaining the goal. Self-efficacy enhances commitment as the employee persists in the task, believing that he can attain it. An employee’s self-efficacy can be increased in several ways by a supervisor; (i) providing suitable training, (ii) finding role-models for the employee to identify with, (iii) persuasive messages that express confidence in the person’s ability – transformational leaders, such as Sir Alex Ferguson, are especially skilled at this, inspiring and stimulating their subordinates. The simple action of setting an employee a difficult goal raises their self-

27 efficacy as it is an implicit expression of confidence from a leader, that he believes in the employee’s ability to attain the goal. Commitment can also be enhanced by convincing the employee of the importance of attaining the goal, this can be achieved in several ways; (i) getting the employee to commit publicly to the goal, (ii) providing them with support along with an inspiring vision, (iii) allowing the employee to partake in the process of establishing the goal – this leads to better performance and a further increase in their self-efficacy.

1.2.4.3.2 Employees’ Self-Regulation

Self-regulation is an important skill in life, especially in stress-intensive environments like the workplace, and goal-setting is a key factor of self-regulation (Locke & Latham, 2002).

By providing employees different training in self-regulation, employers can further increase their employees’ self-efficacy as they enhance their ability to exercise influence over their own behavior in the long-term. Goal-setting research has led to the development of a high-performance cycle; setting challenging goals consequently leads to greater performance, which then leads to rewards – monetary, recognition, or promotion. These rewards produce greater employee satisfaction and enhanced self- efficacy as their perceived ability to meet future challenges grows, along with their confidence at overcoming even more difficult goals.

1.2.4.3.3 The ‘Success-Breeds-Success’ Cycle in Sport

The fundamental principle of the psychological momentum hypothesis hypothesizes a relationship between consecutive performances such that success leads to further success and failure to further failure. Success enhances an athlete’s perception of themselves by enhancing their self-efficacy and motivates them for further success, thus supplying them with psychological momentum.

28 Psychological momentum is defined as “an added or gained psychological power that changes a person’s view of him/herself or of others, or others’ views of him/her and themselves” (Iso-Ahola & Mobily, 1980). This definition suggests that a psychological momentum advantage can be gained or maintained either by an athlete’s own successful performance, or by a competitor’s unsuccessful performance. Based on this definition, recent research has been carried out to try to model these effects (Iso-Ahola & Dotson,

2014). Their investigation of psychological momentum included both “within” a performance, i.e. within a tennis match, and “between” performances, i.e. a day-to-day tennis competition. Based on this definition and its disposition, the researchers made three predictions; “an individual or team that has (a) more momentums during the entire contest is more likely to win or be successful (frequency effect), (b) the one whose momentum lasts longer (duration effect) is more likely to win or be successful, (c) momentums with higher intensity increase the likelihood of success and winning

(intensity effect)” (Iso-Ahola & Dotson, 2014).

1.2.4.3.3.1 The Basic Model of Psychological Momentum

Psychological momentum is grounded on two perceptions of oneself as a performer; (a) without competitive comparison, (b) relative to an opponent. The combination of these two perceptions determines a third – “perceived likelihood of winning or being successful in achieving a future goal” (Iso-Ahola & Dotson, 2014). Therefore, for psychological momentum to be present and experienced, athletes must: (i) have a strong sense of self- efficacy in themselves as performers, (ii) perceive themselves as superior to their opponents, (iii) perceive an increased likelihood of winning or being successful. “Simply stated, a competitor who has gained momentum believes he or she is the cause of his or her success, is highly confident, thinks her or she is better than the opponent, and consequently senses a good probability of winning the contest or achieving his or her

29 stated goals(s). These perceptions combine to form momentum that becomes a psychological force that carries the competitor to further success” (Iso-Ahola & Dotson,

2014). These perceptions describe the basic model of success-breeds-success but are still insufficient without the original catalyst of success nor without preceding success in the previous contest.

1.2.4.3.3.2 A Complex Model of Psychological Momentum

An alternative model included an ‘intensity effect’, ‘frequency effect’, along with a

‘duration effect’. The intensity effect was included based on the hypothesis that psychological momentum and the likelihood of subsequent success would be greater given a greater initial success. For example, a forward scoring three goals in the initial soccer match creates a stronger sense of psychological momentum than scoring only one goal in the initial match. The inclusion of the frequency effect is based on the hypothesis that the the closer in time proximity that the consecutive successes occur, the greater the perceived linkage and thus psychological momentum is more likely to grow. The final effect that the researchers included in the model was the duration effect – following on from the two previous effects, the researcher hypothesised that the longer the chain of successful performances, the stronger the athlete’s sense of psychological momentum.

For example, a stronger sense of psychological momentum occurs after a forward has scored in two consecutive soccer matches in comparison to if he had only scored in the previous match. A period of psychological momentum is terminated either by; (i) a stoppage in time – cutting off the continuity of performance, or (ii) by a performer’s own unsuccessful performance or a successful one by his opponent. The termination of a period of psychological momentum is therefore ultimately unavoidable as it is impossible to avoid human errors in performance.

30 1.2.4.3.3.3 Psychological Momentum as a Mediator and Moderator

Iso-Ahola & Dotson provided two further models that capture the effect of psychological momentum on consecutive successes. The first model depicts the evidence that has been found to support a causal relationship between initial success (푆1) together with psychological momentum on future success (푆2) (see Figure 1). The second model rests on the hypothesis that “initial success (푆1) has two major effects: (a) it increases the likelihood of subsequent success (푆2) in and of itself; and (b) gives rise to psychological momentum and enhances its strength.” (Iso-Ahola & Dotson, 2014. This suggests an interaction effect between the initial success (푆1) and psychological momentum, suggesting a moderation effect (c in Figure 2).

Figure 1

Mediational model of Psychological Momentum (Iso-Ahola & Dotson, 2014)

Figure 2 Moderation model of Psychological Momentum (Iso-Ahola & Dotson, 2014)

31 1.2.4.3.3.4 Objections to the Success-Breeds-Success Model

An examination of the success-breeds-success model (Avugos & Bar-Eli, 2015), suggested that the theoretical framework and conclusions of the research that presented the model was conflicting of empirical research. They argued that the claims of Iso-Ahola & Dotson had only received limited support, calling for more solid empirical research that shows the success-breeds-success principle and the conceptual framework of psychological momentum.

1.2.5 Psychological Momentum and “Hot Hand” Research in Football

Our study set out to discover if the “hot-hand” phenomenon is prevalent empirically for forwards in football. The phenomenon has a theoretical foundation in the sport as evidence of psychological momentum perceptions have been shown to exist for football players, with scoring or conceding goals being the major factors for its emergence. Yet only little research has been carried out in the quest to find the elusive phenomenon. The following is a summary of the research that has been carried out in the field.

1.2.5.1 Footballers’ Perceptions of Psychological Momentum

Jones and Harwood (2008) carried out qualitative research hoping to identify and examine perceptions of psychological momentum during games. Five university football players from a premier U.K. sporting university were selected and interviewed, two females and three males; one defender, two midfielders, and two forwards. Multiple, in- depth interviews were carried out in different time points as the researchers hoped to generate new interview strategies to strengthen the quality of the concepts developed from the analysis.

The triggers of positive psychological momentum included their opponent’s weaknesses or mistakes, good luck, team cohesion, encouragement from teammates, coaches, fans,

32 and scoring goals. The outcomes of this positive momentum included increased confidence, perceived success, and feelings of invincibility. Examples of triggers of negative psychological momentum included the opponent’s strengths and reputation, bad luck, loss of concentration, pressure, nerves and anxiety, fatigue, and conceding goals. While the consequences of the negative momentum included reduced confidence and feelings of threat, frustration and anger, disappointment, hopelessness, and negative body language (Jones & Harwood, 2008).

Redwood-Brown et al. (2017) developed the questions from Jones & Harwood’s (2008) study. They used a mixed methodology, incorporating quantitative research along with qualitative study to investigate the categories and strength of the different triggers and consequences of psychological momentum in football. 75 participants were recruited, including ten English Under-18 League One academy players (six defenders, three midfield players, and one striker), along with four anonymous professional teams from the English football leagues. Data was collected for three sessions, over six weeks, with two weeks between interviews and focus groups.

Categories of positive psychological momentum included general emotions (e.g. positive feelings and low perception of pressure), confidence (e.g. positive attitude and body language), external influences (e.g. encouragement, manager/captain, crowd, opponents), performance factors (e.g. individual performance, effort, and scoring). “All players interviewed associated experiences of positive momentum with scoring a goal”

(Redwood-Brown, Sunderland, Minniti, & O’Donoghue, 2017), even though not all the subjects were forwards. The dimensions of negative psychological momentum consisted of confidence issues (i.e. loss of confidence), group experiences of negativity (e.g. negative team performance, negativity within the team), psychological aspects (e.g. negative body language, feelings and emotions).

33 The research found that goals, whether scoring or conceding them, produced the highest average scores from players’ questionnaire responses for positive and negative PM. “All players interviewed referred to goal scoring as a factor related to positive PM, reporting that a positive PM state would often result in progression toward a goal/scoring and that progressing toward a goal/scoring also triggered positive PM” ” (Redwood-Brown,

Sunderland, Minniti, & O’Donoghue, 2017).

1.2.5.2 The “Hot-Foot” in Football

In footballing terms, the closest concept that is similar to the “hot-hand” is the collective perception that footballers show streaks in their goal scoring performance – the “hot- foot”. Yet, compared to basketball, football is a low-scoring game and thus can only be analysed for momentum between successive games. A study in 2008 did this on a small scale, “treating successive games as successive scoring attempts to investigate whether scoring streaks across sequences of games exist” (Ayton & Braennberg, 2008).

To verify that the belief exists in football, a small survey of fifteen English Premier League players (ten of them international players) were asked two questions regarding their belief in the sequential dependency of scoring. The first question; “Do you think players go on and off form even when fully match fit?”, was answered positively by fourteen out of the fifteen players. The second question; “Imagine one of your players (teammates) had scored in each of his last two games. Do you think he would be more or less likely than usual to score in the next game?”, 13 of the subjects replied “more likely”. These results showed that the “hot-foot” belief is prevalent in most of the small sample of professional players.

Forwards in football are fighting for a limited number of spots in the starting line-up; teams usually line up with no more than three forward players, and sometimes even only

34 employ one. These players are therefore often picked, and dropped, from the starting line-up based upon their recent scoring return – whether they are providing the goals themselves or supplying others with assists for goals. These two stats are usually the major factor when players are thought to be in or out of form. Thus, not scoring or not being part of the build-up of the team’s goals can be a source of pressure to perform and will often be the major influence on a forward’s level of self-efficacy and psychological momentum.

The study researched the hypothesis of the “hot-foot” – do forwards have an increased likelihood to score in a match after scoring in the previous one. Goal-scoring data for the top 12 goalscorers in the English Premier League for the 1994-1995 and 1995-1996 seasons were collected and analysed. Alan Shearer, a retired English Forward and the all- time greatest goalscorer in the English Premier League with 260 goals over his career, was used as an example for the analysis. His records showed no significant evidence for the existence of the “hot-foot” phenomenon and showed that he had a higher likelihood of scoring after not scoring in the previous match than if he had scored in the previous match. The researchers suggested that they had not taken the home advantage (Adler &

Adler, 1978; Pollard, 1986; Nevill & Holder, 1999) into account and, as it has been extensively shown that scoring is more likely in home matches, decided to carry out the analysis twice – for home and for away games separately.

Once again Alan Shearer was their example for the analysis. Shearer had scored in 79% of his team’s home games. This increased to 85% if he had not scored in the previous match and fell to 74% if he had scored – this difference was not statistically significant, but still pointed to conclusions that were contrary to the “hot-foot” phenomenon. Similar results were shown for his away matches. The same analysis was performed for the twelve sample forwards, yet none of them showed statistically significant evidence of the “hot-

35 foot” phenomenon. To fight against a lack of statistical power in each of the twelve separate tests, the data was aggregated and analysed together. Once again, though no statistically significant results were found, home or away, and led the researchers to state that “any belief in the ‘hot-foot’ also appears to be a fallacy” (Ayton & Braennberg, 2008).

1.3 Research Outline

The purpose of this research was to use a vast amount of data, along with more advanced statistical techniques than the ones that have been performed in previous studies, to investigate whether momentum exists in football. We looked at historical data and tested this question from two different perspectives: team-level and player-level.

For the team-level research, we explored to see if previous success in terms of results leads to further successful results in the future. While for the player-level analysis we investigated the ‘hot-foot’ effect amongst strikers in the elite European football leagues.

The research also considered the players’ level of expertise in the analysis; investigating whether the magnitude of the effect of the “Hot-Foot” phenomenon gets weaker as the players’ level of expertise increases.

Redwood-Brown et al. (2017) showed that all the football players they interviewed had associated experiences of positive momentum with scoring a goal, even though not all the interviewees were forwards. This research suggests that this reliance on goals for momentum must be even stronger for forwards whose job it is to score or create goals if they want to keep their place in the team.

For the player-level research, we continued in the same direction as the one employed by

Ayton & Braennberg (2008); considering successive football games as successive scoring opportunities (due to the fact that football is a low-scoring game) and considering the advantage of playing at home and its effects on the likelihood of scoring. The first

36 hypothesis is based on the theory that success (scoring a goal or assisting it) will increase a player’s chances of success in their next game. The second hypothesis is based on the

Bandura’s theories of self-efficacy – as a person matures: (i) self-efficacy stabilizes

(Bandura, 1994), (ii) the elite forwards gain and accrue success experiences over their career (performance accomplishments – Bandura 1977) by playing, scoring, and assisting,

As the performance accomplishments accrue over an elite forward’s career, their reliance on self-efficacy along with the effects of failures are hypothesised to decrease.

1.3.1 Research Questions

1.3.1.1 Team-Level i. Do recent, previous successes influence a team’s probability for future success?

1.3.1.2 Player-Level i. Do recent, previous successes influence the probability for future success for elite

football forwards? ii. If such an effect exists, is the magnitude of the effect constant throughout a

forward’s career?

1.3.2 Research Scope

Data was collected for teams and for the elite forwards who are playing or have played in any of the top-five European Football Leagues; English Premier League (EPL), Spanish La

Liga (La Liga), German Bundesliga (Bundesliga), Italian Serie A (Serie A), and French Ligue

1 (Ligue 1). The research answers Wardrop’s (1991) call to carry out an analysis over several years. Specifically, each player’s entire career data was collected and analysed – or as far back as the records allowed.

37 1.3.3 Project Outline

The next chapter will explain in detail the data that was collected, and the methodology that was carried out as part of this study. In the following chapter, we will present the results of our observational research and then finally discuss the applications and implications of the results along with future research directions.

38 Chapter 2

Method

Our study’s aim was to carry out an observational research in an attempt to discover whether momentum exists in football from two different perspectives as outlined in the introduction chapter: i. Player-Level

We set out to research how previous performances by a forward, along with additional factors, affect his future performance – specifically, we looked at how previous successes in terms of goal-scoring in the previous match affect his chances of scoring in the next match.

Therefore, for the purpose of researching momentum in individual players, we chose goals as the central indicator of success. Due to the inherent fact that football is a relatively low-scoring sport, we decided to only look at the elite forwards in the game whose main objective is to provide goals and goal-scoring opportunities for their teams.

Therefore, two similar definitions were decided upon for success from a forwards’ perspective:

i) Scoring – the player took the shot that led to the goal

ii) Scoring or Assisting – the player either scored a goal, or provided the pass to

the player that scored the goal directly after receiving the pass

We then tested whether success in an upcoming match was dependant on previous success in their preceding match.

39 ii. Team-Level

In this research level we looked how previous performances, specifically previous successes in terms of results, together with other factors, affect future results. For the purpose of researching momentum in teams we chose to look at results, specifically winning, as the main indicator of success in football. We looked at the effects of a team’s previous results on their probability of success in their next match.

2.1 Data Sources

The raw data for this research was retrieved from the internet from two major sources:

◦ www.tranfermarkt.com

◦ www.elofootball.com

2.2 Data Description

2.2.1 Player-Level

Football is unique from other sports such as basketball and hockey as it is low-scoring. It is the forwards that carry most of the responsibility for providing the goals for their teams, and thus we decided to only research the elite forwards, selecting them based on the number of goals they scored.

The goals scored in a season seem to have a power-law distribution (see Figure 3) where many players score a small amount of goals, and a small number of players score a lot of goals. The players we chose to investigate are the 95th percentile of the scorers – they are predominantly forwards, but some midfielders also made it into the analysis based on the goals-scored criteria.

40 Figure 3 Distribution of Goals Scored per Season in the past 10 Years in the “Big-5” European Football Leagues

The 146 players (see Appendix A) analysed had scored more than the 95th percentile of goalscorers in their league (see Table 1) in at least two different seasons from 2008-2009 to 2018-2019.

Table 1 The 95th Percentile of goals scored by a player in the “Big-5” Leagues League 95th Percentile of Goals Scored in a Season

English Premier League 12 Goals

Spanish La Liga 13 Goals

Italian Serie A 12 Goals

German Bundesliga 12 Goals

French Ligue 1 11 Goals

Furthermore, data was also collected for 36 legendary forward players (see Appendix B) to investigate whether the momentum phenomenon was prevalent in their careers.

41 In total we had collected 182 players’ data, spanning the entirety of each player’s career – where each row in the data represents a match that the player’s team participated in whether the player participated in the match or not. Overall, data was collected for

125,109 matches from 2000 to 2019 with 18 data fields (see Table 2).

Table 2 Raw Data Variables for Player-Level Analysis Data Datum Data Field Description Type Example

Name Player’s name TEXT Eden Hazard

Date of Birth Player’s date of birth TEXT 1991-01-07

Season in which the match was Season TEXT 2007/2008 played

Competition in which the match Competition TEXT Ligue 1 was played

Round of Competition that the Round TEXT 15 match was played in

Date Date of the match DATE 2007-11-24

Venue of the match, from player’s Venue TEXT Away point of view – Home/Away

Team Played For Team that player represented TEXT LOSC Lille

AS Nancy- Opponents Opponents played against TEXT Lorraine

Score The final score of the match TEXT 2:0

The main position that the player Position TEXT Forward played in during the match

Goals How many goals the player scored INTEGER 0

42 How many assists the player Assists INTEGER 0 provided

The minute that the player Yellow Card TEXT 0 received a yellow card (if he did)

The minute that the player Second Yellow Card received a second yellow card (if TEXT 0 he did)

The minute that the player Red Card TEXT 0 received a red card (if he did)

How many minutes the player Minutes Played INTEGER 12 played for during the match

Reason for not The reason that the player didn’t TEXT - Playing play in the match

2.2.2 Team-Level

For the team-level analysis, raw data was collected for all of the teams that had played in the Big-5 European football leagues in each of the seasons; from the 2000-2001 season to the 2018-2019 season.

The sample consisted of 206 teams (see Appendix C) from 5 different countries, with each team’s match history for the nineteen-year-period collected. Each row in the data represents a match in which the team had participated. A total of 161,636 matches from

98 different competitions with 13 data fields (see Table 3).

43 Table 3

Raw Data Variables for Team-Level Analysis

Data Field Description Data Type Datum Example

Team Name Team’s name TEXT Chelsea FC

Season in which the match was Season TEXT 2017/2018 played

Competition in which the match UEFA Champions Competition TEXT was played League

Round of Competition that the Round TEXT Group C match was played in

Date Date of the match DATE 2017-10-18

Time Time of the match TIME 19:45

Home Team Name of the hosting team TEXT Chelsea FC

Away Team Name of the visiting team TEXT AS Roma

Main formation used by the Formation TEXT 3-5-2 team during the match

The tactical intent of the Formation Type TEXT Flat formation

Manager Name of the team’s manager TEXT Antonio Conte

The number of people present Attendance INTEGER 41,105 at the match

Result The final score of the match TEXT 3:3

44 2.3 Feature Engineering

Feature engineering is a crucial step in the machine learning process, it enables the construction of new variables through the transformation of the collected data in addition to incorporation of domain knowledge. During the process we engineered both the dependant variables as well as the independent variables to provide context to the data.

2.3.1 Player-Level Feature Engineering

Since we were attempting to discover the factors which influence a player’s chances of providing goals, the minimal requirement was that the player had participated in a match.

Therefore, following the feature engineering process, all the data rows from matches in which the players did not participate were removed – leaving us with 88,029 match data rows. The raw data collected was from 213 different competitions, with matches from as early as June 5th, 1965 up until the 27th of July 2019 (see Figure 4). Each player was given a unique id, as was each season.

Figure 4 Distribution of Seasons in which the Players had Played

45 2.3.1.1 Dependent Variables

Two related dependent variables (see Table 4) were constructed to measure success and test whether a momentum phenomenon is prevalent for forwards, both centred around the forwards’ fundamental purpose in the game – providing the goals for their team.

The first variable indicated whether the player had scored or not during the match, while the second variable was a more relaxed definition of success – asserting that the player was successful in a match if they had either scored themselves or provided assists for teammates during a match. The players in our sample had scored in 32% of the matches they had appeared in (see Figure 5), while they had scored or assisted in just over 40% of the matches (see Figure 6).

Table 4 Success Variables for Player-Level Analysis Data Field Description Data Type Example

Whether the player had scored during Scored BOOLEAN False the match

Whether the player had scored or Scored/Assisted BOOLEAN True provided an assist during the match

Figure 5 Distribution of Matches in Players had Scored

46 Figure 6 Distribution of Matches in Players had Scored or Assisted

2.3.1.2 Independent Variables

2.3.1.2.1 Experience

In the attempt of creating a variable that best measures a player’s experience level, three different yet related features were engineered (see Table 5).

Table 5 Experience Variables for Player-Level Analysis Data Field Description Data Type Example

The age of the player at the date of the Age FLOAT 21.2 match

The age of the player at the date of the Age2 FLOAT 449.44 match squared How many minutes the player had Accrued accumulated during his entire career INTEGER 3,803 Minutes prior to the match

How many appearances the player had Accrued featured in during his entire career INTEGER 109 Appearances prior to the match

47 Age was finally chosen as the optimal proxy variable for experience, as the two other alternatives were incomplete for some of the players who did not have complete data for the beginning of their careers. Therefore, we considered Age to be a suitable proxy for experience due to its strong positive correlation with the two other alternatives; 0.86 with Accrued Minutes, and 0.88 with Accrued Appearances respectively. Age squared was also calculated as we suspected a nonlinear relationship between Age and Success

Figure 7

Distribution of Player’s Age

2.3.1.2.2 Player Recovery Time

It is well established in the sports-science field that recovery is a crucial aspect of an athlete’s routine as part of his professional-athlete lifestyle. We therefore decided to include recovery as an independent variable, measured as the number of days that the player had to rest and recover from his previous appearance. This time could be very short when teams’ schedules towards the end of a season can often include three matches in the space of 7-8 days, depending on the number of tournaments that they are competing in simultaneously.

48 This time can also be very long when players have long lay-offs from playing in order to recover from an injury or when they are on holiday from work during the summer months and winter months in some leagues. We considered these instances to be outliers in the data and they were removed, as we set the maximal rest time to be 21 days, leaving us with 82,327 rows of players’ match data.

Table 6 Recovery Variable for Player-Level Analysis Data Field Description Data Type Example

The number of days between Rest Time INTEGER 7 consecutive appearances by a player

Figure 8 Distribution of Players’ Rest Time Between Appearances

2.3.1.2.3 Minutes Played

Up to three players can be substituted during the 90 minutes of a football match.

Therefore, players were assumed to be substitutes in a match if they had played less than the full 90 minutes. It is intuitive that the more time that a player is given to play, the higher the likelihood of him scorings. There were 664 match rows in which a player had played more than 90 minutes, these were matches that went into extra-time and

49 considered outliers in the data that could skew the results of the modelling. As such, it was decided to remove them, leaving us with 81,663 data rows.

Table 7 Minutes Played Variable for Player-Level Analysis Data Field Description Data Type Example

How many minutes the player Minutes Played INTEGER 90 had played during the match

Figure 9

Distribution of Minutes Played

2.3.1.2.4 Player’s Status in Team

A player is more likely to possess higher match fitness and be less mentally stressed about having to earn his place in the starting line-up if he is chosen to play more frequently. We created two variables to quantify this. Firstly, a variable that indicates whether the player had played in his team’s previous match or not (see Figure 10). The second variable measures the magnitude of the players’ importance to the team – a count of how many consecutive matches the player has played prior to the current match (see Figure 11). In the construction of this variable we ignored fatigue and assumed that it doesn’t become a factor, as the player would not be given match-time by his coach if he were not fully fit.

50 Table 8 Player’s Status Variables for Player-Level Analysis Data Field Description Data Type Example

Whether or not the player played Played Previously BOOLEAN True in the team’s previous match

In how many consecutive Consecutive matches the player has INTEGER 5 Appearances participated prior to the match

Figure 10

Distribution of Matches in which the Player had Played Previously

Figure 11

Distribution of Consecutive Appearances by a Player

51 2.3.1.2.5 Previous Success

The variables of major interest to our research question deal with the momentum which the forward player possesses coming into the match, as a result of providing goals in their previous appearance(s). Once more, we have created two variables that attempt to quantify this momentum (see Table 9). These are based on each of the two varying success criteria; (i) scoring a goal or (ii) being involved in a goal by either scoring or assisting.

The first variable indicates whether the player had scored in his team’s previous match or not. While the second variable (see Table 10) is a measure of the magnitude of the players’ momentum or streak of momentum – a count of how many consecutive matches the player has either scored in or scored/assisted in.

By feeding theses previous success variables into our model, we will be able to verify and quantify the effects of a forward’s previous success on his likelihood of success in the match following directly after. These variables will be the determining variables in our exploration into the existence of momentum on a player-level in football forwards.

Table 9 Previous Success Variables for Player-Level Analysis Data Field Description Data Type Example

Whether or not the player had Scored Previously scored in his team’s previous BOOLEAN True match

Whether or not the player had Scored/Assisted scored or assisted in his team’s BOOLEAN False Previously previous match

52 Figure 12 Distribution of Scoring in Previous Appearance

Figure 13 Distribution of Scoring or Assisting in Previous Appearance

Table 10

Magnitude of Previous Success Variables for Player-Level Analysis

Data Field Description Data Type Example

In how many consecutive matches the Scoring Run INTEGER 3 player has scored prior to the match

In how many consecutive matches the Scoring/Assisting player has scored or assisted prior to INTEGER 5 Run the match

53 Figure 14 Distribution of Scoring Run

Figure 15 Distribution of Distribution of Scoring/Assisting Run

2.3.1.2.6 Home Advantage

Evidence of home-field advantage has been found in many sports, and football is no different. The support of the majority of the crowd in attendance, together with the familiarity of the field and the minimal travel time are some of the factors that come into play when playing at home. These factors help teams produce better performances and better results at home compared to playing away, and likewise potentially improve a forwards’ chances of scoring or assisting during the match.

54 Table 11

Home Match Variables for Player-Level Analysis

Data Field Description Data Type Example

Whether or not the player’s team was Home playing the match at their home BOOLEAN TRUE stadium

Figure 16 Distribution of Match Venues

2.3.2 Team-Level Feature Engineering

For the team-level, we engineered the factors which influence a team’s chances of winning a match in their country’s top league. League matches are often referred to as the “bread and butter” of a team’s match schedule – demonstrating that the league is the most important and fundamental part of the season. Therefore, an important consideration for the research was to investigate whether matches from other competitions were of different importance for the teams. Thus, variables were engineered once with consideration of all the teams’ matches – irrelevant of the competition that the match was in, and secondly after filtering the data so that only matches from the top league in each respective country were considered. Thereafter, following the feature engineering process, all the data rows from matches in both

55 domestic and international cup competitions were removed along with the match rows from the lower leagues in the country, leaving us with 68,500 data rows of matches.

The data after the engineering stage included match information from the Big-5 Leagues, with matches from as early as July 28th, 2000 and up until the 26th of May 2019 (see

Figure 17). There are 3,652 match data rows for every season except for the first four seasons – due to some leagues consisting of less teams in those seasons. The English

Premier League and the Spanish La Liga contained 20 teams for the entire analysis period.

The French Ligue 1 used to contain 18 teams before it was expanded to 20 teams in the

2002/2003 season. While the Italian Serie A contained 18 teams until the 2004/2005 season when it was expanded to 20 teams. Each team was given a unique id, as was each season.

Figure 17

Distribution of Seasons in Team-Level Analysis

2.3.2.1 Dependent Variable

The definition of a team’s success in a match is dependent on the result that they produce at the end of it. For the purpose of this research – we only considered a win to be a success. Therefore, either losses or draws were considered failures. The teams in our

56 sample had won in almost 37% of the matches they had appeared in (see Figures 18 and

19).

Table 12 Success Variables for Team-Level Analysis Data Field Description Data Type Example

Won Whether or not the team won the match BOOLEAN False

Figure 18

Distribution of Results

Figure 19

Distribution of Success in terms of Wins

57 2.3.2.2 Independent Variables

2.3.2.2.1 Team Quality and Opponent Quality

A major factor in a competitive sporting event’s outcome is the quality of each participant, and football is no different. Yet, quantifying a teams’ sporting ability is a major challenge. For this purpose, we decided to use the well-established Elo Ratings system originally invented by Arpad Elo to quantify relative skill levels of chess players.

The Elo ratings system is based on a zero-sum game in which the “winner-takes-all”. The system’s mechanism is dynamic, updating each team’s ranking after every contest through the exchange of Elo points between the participants based on the outcome of the contest, and their relative Elo ratings prior to the contest (see Table 13).

The power of the system comes from its ability to predict the outcome of contests, calculated by using a logistic function (see Figure 20) and using on the difference in the

Elo ratings between the two competitors as the input. For example, a pre-match Elo

Ratings difference of zero translates into a 50% chance for each competitor to win (with the probability of a draw ignored).

Table 13 The Elo Ratings Mechanism

푫푹 = 퐸퐿푂푇푒푎푚 − 퐸퐿푂푂푝푝표푠푖푡푖표푛 (1)

DR: the difference in Elo ratings between the team and its opponents prior to their contest

1 (2) 푷(푾풊풏푻풆풂풎) = 퐷푅 − 10 400 + 1

(3) 푷(푾풊풏풐풑풑풐풔풊풕풊풐풏) = 1 − 푃(푊𝑖푛푇푒푎푚)

58 Each team’s probability of winning the contest is calculated based on their Elo ratings difference.

A DR of 0 (implying that two teams are of identical quality) equates to each team having a 50% probability to win the contest.

휟 푬푳푶푻풆풂풎 = 푘 ∗ 퐺퐷푀 ∗ (푅푒푠푢푙푡 − 푃(푊𝑖푛푇푒푎푚)) (4)

휟 푬푳푶푶풑풑풐풔풊풕풊풐풏 = −휟 푬푳푶푻풆풂풎 (5)

푅푒푠푢푙푡 = 1, 𝑖푓 푊표푛

푅푒푠푢푙푡푠 0.5, 𝑖푓 퐷푟푒푤 (6) 푅푒푠푢푙푡푠 0, 𝑖푓 퐿표푠푡

Each team’s rating is updated post-match and depending on the result.

GDM: the goal-difference multiplier – Causes larger ratings exchanges for bigger wins (losses).

K: update volatility factor; determines the “sensitivity” of how wins and losses impact the Elo Update magnitude. Larger k values increase the volatility of the ratings over time, while smaller k values lead to stable ratings.

Figure 20 Elo-Difference Logistic Curve

59 Elo based ranking systems are commonplace in many sports including basketball and baseball. Its validity in the football world is demonstrated through the adoption of a simplified Elo system by FIFA in 2018 for its current version of national teams’ rankings.

Table 14 Elo Ranking Variables Data Field Description Data Type Example

The difference in Elo Ratings between the Elo Difference INTEGER 55 team and its opposition

Team Elo The team’s Elo Rating INTEGER 1978

Opposition Elo The opposition’s Elo Rating INTEGER 1923

Figure 21 Elo-Difference Distribution

2.3.2.2.2 European Competition Commitments

As mentioned previously, league matches are usually considered as the fundamental part of the teams’ schedule. Yet, a small number of the best teams from the previous season

(usually between 4-6 teams) in each league get the opportunity to participate in continental competitions such as the UEFA Europa League (previously UEFA Cup), or the more prestigious UEFA Champions League. The fixtures in these competitions are played during the middle of the week, with teams often needing to travel a substantial distance

60 to the opponent’s country for the away matches, while the rest of the teams in the league

can rest and prepare for their upcoming matches in the league.

These international competitions are prestigious in terms of broadcast revenue for the

participating teams, the possible prize money for making it into the later stages of the

competitions, and the priceless exposure to the international football audience and the

resulting potential marketing opportunities. It is not uncommon for teams, especially

those who make it into the later stages of these competitions, to place greater emphasis

on these matches and to rest some of their star players in the league matches before or

after these continental matches. Therefore, we engineered two variables that indicate

whether the team had a match in a European competition before or after their league

match (see Figures 22 and 23).

Table 15 European Competition Participation Variables Data Field Description Data Type Example

Europe Indicates that the team’s previous match was BOOLEAN False Before in a European competition

Indicates that the team’s next match was in a Europe After BOOLEAN True European competition

61 Figure 22 Distribution of Matches with European Fixture Directly Before

Figure 23 Distribution of Matches with European Fixture Directly After

2.3.2.2.3 Home Advantage

Similarly to the player-level analysis, possessing the home advantage is a factor of the

probability of winning a contest. As the name alludes to – a team is expected to, on

average, have a certain advantage over its opponents when playing at their home ground

compared to playing them away from home. We therefore created a variable that

indicates whether the team was playing at home or not for each match (see Figure 24).

62 Table 16 Home Advantage Variable in Team-Level Analysis Data Field Description Data Type Example

Indicates whether the team was playing Home Team BOOLEAN True at home or not

Figure 24 Distribution of Home Advantage in Team-Level Analysis

2.3.2.2.4 Change of Manager

The football manager’s job today is risky and unstable, with the likelihood of keeping their position mainly a function of the results that their teams produce.

A winning team is a happy, united group of players and staff, along with joyful fans who look forward to each match with great optimism. A losing team that is playing badly though will quickly turn into a gloomy situation as players often fall-out with one another or the staff, with the blame for the poor performances and results cast upon one another. Fans often become restless and begin calling for the people responsible to take action and be accountable for their team’s poor results before things get out of hand in terms of failing to meet expectations. Yet, the main burden of responsibility for a team’s results will almost always fall on the shoulders of the manager.

63 The manager typically picks the players for the team, chooses the match strategies and tactics, the training regimen and is by and large responsible for the group spirit within the changing rooms of football teams. When results do not meet expectation, the first and often only step that can be taken by the decision makers at board levels of football clubs is to dismiss the manager of his duties and replace him. Several academic studies have found evidence for a short-term improvement to results following managerial changes, and we incorporate this into our research in the form of a variable that indicates that a manager change has occurred. i.e. A manager’s contract with the football team was terminated, and a replacement employed.

Table 17 Managerial Change Variable Data Field Description Data Type Example

Indicates whether the manager was Manager Change replaced between the previous match BOOLEAN True and the current match

Figure 25 Distribution of Manager Change

64 2.3.2.2.5 Historical Results

The history of league results between teams can provide new information that cannot be captured by our other variables. Some teams and venues simply provide greater challenges to certain teams compared to other teams of similar quality due to other factors.

The classic example is a local derby, where two teams from the same city or area face each other in a match. The contest is often more emotionally charged than other fixtures, as friends, family, and co-workers become bitter enemies for a short time before the match kicks off. These matches are often just as emotional on the pitch, with players drawing on the passion stemming out from the stands as their own fans and their rival’s fans cheer their respectable team on.

These matches are often the first fixture that the fans look out for when their team’s fixture schedule is first published before the season has begun. The results of such local derbies, competitive derbies between teams vying for a similar place in the league table or matches against “bogie teams” can often be surprising. Thus, leading to the old cliché adored by commentators – where the two teams’ form coming into such a match can be

“thrown out of the window”.

We have created a variable that looks at the previous results of the same fixture, with the location of the match considered. i.e. the same team hosting the opposition or vice-versa.

The variable is calculated as the average points that the team has produced historically from the fixture from up to four previous meetings – under the logic that teams’ quality gap is not expected to switch directions between successive meetings and therefore the history of a matchup’s results contains information that cannot be obtained from our other variables.

65 Table 18 Historical Matchup Result Variable Data Field Description Data Type Example

The average number of points that the team Match History has historically produced from the fixture FLOAT 1.2 from the previous 4 encounters

Figure 26 Distribution of Average Points from Match Historically

2.3.2.2.6 Match Congestion and Recovery Time

Again, in a similar fashion to rest time between games for the Player-Level analysis, the match schedule congestion and related recovery time is crucial. Teams that have high congestion and thus a small recovery time are more likely to rotate the players in their squad and rest important players to avoid fatigue in the team potential injuries.

Teams that only participate in league matches, will usually have around 7 days of rest during most of the season (see Figure 27), up until the final rounds of the league. Yet, teams who qualify for the later stages of the domestic cup competitions will often have cup games during the middle of the week, and thus three or four days between matches

(and the possibility of playing up to 120 minutes instead of 90 in the case of a draw at the

66 end of the 90 minutes). This congestion gets even heavier if you are a top tier team who is also competing on the European front, again with midweek matches – and even worse again if the team had to earn its qualification to the European competition in the early part of the season.

It is not uncommon to find teams that overperformed in one season, earning a spot in one of next season’s European competitions, only to struggle to replicate the same performances and results in the following season when they are juggling the league schedule as well as the continental one.

Our variable for rest time is calculated as the number of hours that have passed between the kick-off of the current match and the kick-off of their previous match. The same was calculated for the time difference between the current league match and the previous league match.

Table 19 Rest Time Variable for Team-Level Analysis Data Field Description Data Type Example

The number of hours between the kick-off of Rest Time the current league match and the kick-off of FLOAT 160.5 the previous match from any competition

The number of hours between the kick-off of League Rest the current league match and the kick-off of FLOAT 143.0 Time the previous league match

Figure 27 Distribution of Rest Time Between Successive League Matches

Figure 28 Distribution of Rest Time Between Successive Matches

Outlier rest times were removed – defined as any matches with rest time greater than 16 days (384 hours) and lower than 2 days (48 hours) were removed as suspicious data. Data points with rest time greater than 16 days can occur as every league analysed, except for the English Premier League, has a winter break from mid-December until the end of

January or start of February which can last up to 50 days (1200 hours). As this happens for only one round during a season, and dramatically increases the domain of this variable, it was decided to exclude these matches from the data set.

68 2.3.2.2.7 Previous Success – Won Previous Match

In an analogy to the Player-Level analysis, in the team-level analysis we wanted to research how previous wins affect the probability of winning the next match.

Therefore, we created a momentum indicator variable that indicates whether the team had won its previous match directly before the current match. Once again, this variable was broken down into two variables; (i) looking at the previous match irrespective of the competition that it was in, (ii) only looking at the result of the previous league match due to teams emphasising their league matches as mentioned before.

By including this variable as an independent variable in our model, we could verify and quantify the effects of previous success in terms of winning on the likelihood of winning the following league match directly after – and to test whether a momentum effect is prevalent on a team-level.

Table 20 Previous Success Variables for Team-Level Analysis Data Field Description Data Type Example

Won Previous Whether the team won its previous BOOLEAN True Match match, irrespective of the competition

Won Previous Whether the team won its previous league BOOLEAN False League Match match

Figure 29 Distribution of Won Previous Match

Figure 30 Distribution of Won Previous League Match

This variable was engineered to be the short-term-memory momentum variable for the team-level analysis. Next, we considered a more long-term-memory momentum variable which remembers the results from more than one match.

2.3.2.2.8 Previous Success – Form

Results in football translate into points which are accrued during a season. The teams are then ranked in the league according to the points they have accumulated – where the first placed team is the one with the most points, and the last placed teams has the least

70 points . We considered the long-term momentum that the team possessed prior to a match to be affected by their recent results, and not only based on the result of their one previous match.

Specifically, we looked at the average points that they had picked up in their last 4 fixtures as the measure of their momentum. This was calculated in two different ways; (i) the average points picked up by the team in their last 4 matches, (ii) a weighted average of the points picked up by the team in their last 4 matches (where the match closer in time proximity receives roughly 50% more importance than the previous). The weighted average symbolizes an assumption that the more recent the result, the fresher it is in the teams’ mind and thus has more of an influence on their momentum coming into in the next match. These variables will also be used to test whether a momentum effect exists on a team-level in football.

Table 21

Calculations for Form Variables in Team-Level Analysis

4 1 푨풗풆풓풂품풆 푷풐풊풏풕풔 = ∑ 푃표𝑖푛푡푠 (7) 풕 4 푡−푖 푖=1

The average points picked by the team from their last 4 matches

(8) 푾풆풊품풉풕풆풅 푨풗풆풓풂품풆 푷풐풊풏풕풔풕 = 0.12 ∗ 푃표𝑖푛푡푠푡−4 + 0.18 ∗ 푃표𝑖푛푡푠푡−3 + 0.28 ∗

푃표𝑖푛푡푠푡−2 + 0.42 ∗ 푃표𝑖푛푡푠푡−1

The weighted average points picked by the team from their last 4 matches

For both types of averages, an average of 3 points indicates that the team had won its last four matches, an average of 0 indicates that the team lost all of its last four matches, while any number in between indicates mixed results. The two Average Points variables were calculated twice; (i) considering all the teams’ match results irrelevant of the

71 competition that they were in (see Figures 31 and 32), (ii) only including the teams’ league results(see Figures 35 and 36).

Table 22 Form Variables in Team-Level Analysis Data Field Description Data Type Example

The average number of points that the team Average Points FLOAT 2.4 has attained from its last 4 matches

Weighted A weighted average number of points that FLOAT 2.7 Average Points the team has attained from its last 4 matches

Figure 31

Distribution of the Average Points

72 Figure 32

Distribution of the Weighted Average Points

Table 23 Form Variables in Team-Level Analysis Data Field Description Data Type Example

Average League The average number of points that the team FLOAT 2.4 Points has attained from its last 4 league matches

Weighted A weighted average number of points that Average League the team has attained from its last 4 league FLOAT 2.7 Points matches

Figure 33

Distribution of the Average League Points

73 Figure 34

Distribution of the Weighted Average League Points

2.4 Prepared Data

After the data engineering stage and cleaning of the data, we were left with 81,663 rows of players’ match data and 64,776 rows of teams’ match data for statistical modelling.

2.5 Statistical Model

For both levels, a mixed-effects logistic regression was chosen to carry out the analysis as it facilitates us with two important characteristics. First, this model enabled us to see the probabilistic effects of the different factors of interest on the chances of success for each of our analysis levels. This will make the interpretability of the results more intuitive as we will be able to test hypotheses on the factors and measure the magnitude of their effects in terms of probability. Secondly, it provides us with the freedom to separate the effects of the individual or team from the population’s effects which are of interest to us.

The statistical model that we used for both analyses is a member of the Generalized

Linear Mixed Models (GLMM) family. This model was chosen for the following main reasons:

74 i) Binary Dependant Variables

The model is appropriate for the data that we have collected as the dependent variable in logistic regression models is binary (0 or 1), just like the dependent variables in both of our analysis-levels. Our success variables receive a value of 1 if the subject (player; team) was successful (provided a goal; won) and a value of 0 if the subject was unsuccessful

(didn’t provide a goal; didn’t win) ii) Mixed Effects

A mixed effects model contains both fixed effects and random effects. These models are often used within repeated measurements or within-subject studies where measurements are taken for the same subject over time. This is appropriate for our data observations as they are a collection of the same measurements over time of the same subject (player; team) .

A random effect is one that is not repeatable. If the experiment is replicated, then the researchers are expected to estimate different effects but could estimate the variance of the effects from another sample. In our analyses, these effects will be input into the models in the form of unique ID’s for seasons, players, and teams respectively.

A fixed effect is one that is repeatable. These are the population effects and are of interest . They are expected to essentially produce the same covariate regression coefficients if an experiment is replicated. iii) Interpretability

Logistic regressions employ a transformation of the binary dependant variable through a logit transformation. That is, at the end of the data modelling, our model’s parameters can be used to predict the log odds of an outcome. What enables the modelling is the

75 linear relationship between the log odds of the dependant variables with the independent variables. Once that relationship has been modelled, the log odds can easily be transformed into probabilities which are easily interpretable. iv) Base Rates

We will be incorporating the specific players/teams as random effects in the mixed logistic regressions, in terms of the random intercepts and possible random slopes for different independent variables. The random intercepts will enable us to quantify each player’s/team’s different base-rates for scoring/winning, and to see how previous successes affect the probability for future successes relative to their personal and unique base-rate for success.

2.5.1 Model Definition

In GLM Models; 푌 is the dependent variable, observed independently with different values of fixed independent variables 푥1, 푥2, … , 푥푛 and random independent variables

푧1, 푧2, … , 푧푛. The independent variables have a linear relationship to the linear predictor

(9)

The mean µ is a smooth invertible function of the linear predictor η

(10)

This function can then be transformed into a link function. 푔−1 is called the link function, it is the function that links the expected value to the linear predictor.

76 (11)

Specifically, in a mixed effects logistic regression:

The dependent variable is binary and has a Bernoulli distribution.

(푌|풙, 풛 ~ 퐵푒푟(휋(풙, 풛))) (12)

To simplify the notation, we will use 휋(푥) to represent the conditional probability of Y given 풙 and 풛

(13)

The form of the logistic regression that we used is:

(14)

We then applied a link function in form of the logit transformation on 휋(푥), such that we get a linear relationship between the log odds of Y and our fixed and random independent variables:

(15)

(16)

Two things to consider is that there is no error term as we are modelling probabilities, and that a normal distribution is assumed for the random effects N(0, Sz).

Conditionally on the random effects, z, we get that:

풆풙풑(풙휷 + 풛풃) 퐸(푌 | 풛) = 푃(푌 = 1| 풛) = 휋(풙, 풛) = (17) ퟏ + 풆풙풑(풙휷 + 풛풃)

77 And marginally:

퐞퐱 퐩(풙휷 + 풛풃) 퐸(푌) = 푃(푌 = 1) = 휋(풙) = ∫ 풇(풛) 풅풛 (18) ퟏ + 퐞퐱 퐩(풙휷 + 풛풃)

2.5.2 Model Selection

We will be training the models using a backward-stepwise regression. In this method, all the possible independent variables are incorporated in the original model (called the full model), and are removed one by one for as long as the removal of the next variable improves the model in terms of its Akaike Information Criterion (AIC). AIC’s great power comes from its ability to consider the trade-off between the goodness of fit of a model

(similarly to R Squared) and the simplicity of the model in terms of the number of independent variables that it contains.

푨푰푪 = ퟐ풌 − ퟐ 풍풏(푳̂) (19) 푘 − 푛푢푚푏푒푟 표푓 푒푠푡𝑖푚푎푡푒푑 푝푎푟푎푚푒푡푒푟푠 𝑖푛 푡ℎ푒 푚표푑푒푙 퐿̂ − 푡ℎ푒 푚푎푥𝑖푚푢푚 푙𝑖푘푒푙𝑖ℎ표표푑 푓푢푛푐푡𝑖표푛 푓표푟 푡ℎ푒 푚표푑푒푙

After fitting and choosing the optimal models, we will compare the conditional and marginal 푅2 of each model. This comparison will provide us with an understanding of how much the inclusion of the random effects into the model improved its predictability – even though predictability is not the aim of this study. The traditional 푅2 of a fixed effects regression describes the proportion to which a model accounts for the variation of a given data set. In the case of mixed-effects models, the marginal 푅2 describes the proportion of variance that is explained by the fixed factor(s) alone, while the conditional 푅2 describes the proportion of variance explained by both the fixed and random factor(s).

78 Chapter 3

Results

Using a backwards stepwise algorithm on the full models and using the AIC as the decision criteria for optimal model, the following results were obtained after modelling the data :

3.1 Player-Level Analysis

In the player-level analysis we set out to research two hypotheses regarding momentum in football:

➢ Does scoring in the previous match increase the probability of a forward scoring in the

following football match?

➢ Does the forward’s age act as a moderator of the relationship between scoring in the previous

match and the probability of scoring in the following football match?

The research was carried out two times, each with a different measurement of success for a forward:

1. Scoring

2. Scoring and/or Assisting

3.1.1 Scoring a Goal is Considered as Success

Table 24 Measures of Fit: Player-Level Analysis - Scoring a Goal is Considered as Success AIC 96257.0

79 Table 25 Pseudo R-Squared of Player-Level Analysis - Scoring a Goal is Considered as Success Pseudo R-Squared Marginal Conditional 0.099 0.126

Table 26 Random Effects of Player-Level Analysis - Scoring a Goal is Considered as Success Groups Variable Name Variance Std. Dev Correlation Players Intercept 0.104 0.323 Scored Previously 0.011 0.104 -0.35 Seasons Intercept 0.002 0.047 N = 81,663, groups: Players, 182: Seasons, 56

Table 27 Fixed Effects of Player-Level Analysis - Scoring a Goal is Considered as Success Variable Name Estimate Std. Error P Value Intercept -2.850 0.092 < 2e-16 *** Minutes Played 0.022 0.000 < 2e-16 *** Age 0.011 0.003 9.04e-05 *** Age2 -0.004 0.000 < 2e-16 *** Scored Previously 0.308 0.109 0.005 ** Age * Scored Previously -0.009 0.004 0.022 * Scoring Run 0.026 0.015 0.081 ‘ Played Previously -0.013 0.028 0.649 Rest Time -0.003 0.002 0.290 Home 0.377 0.016 < 2e-16 *** Signif codes: 0.001 “ *** ” 0.01 “ ** ” 0.05 “ * ” 0.1 “ ’ ”

The following results were obtained from the optimal model using the model selection method outlined in the previous chapter:

80 The Scored Previously (p < 0.01) variable is statistically significant and a coefficient of

0.348. This suggests that a player scoring in the previous match does increase their probability of scoring in the next match – and suggests that a momentum effect exists for forwards such that scoring in a match increases the likelihood of scoring in the next match.

The interaction variable between Age and Scoring Previously (p < 0.05) is statistically significant and has a coefficient the size of -0.009. This suggests that Age does play a role as a moderator in the effect size of Scoring Previously; as a forward’s age increases and he matures, the momentum effect from scoring in the previous match is weakened (see

Figures 36 and 37).

Figure 36 Probability of Scoring Away by Age and Scoring Previously (Played 90 Minutes) Player-Level Model with Scoring as Dependant Variable

Figure 37

Probability of Scoring at Home by Age and Scoring Previously (Played 90 Minutes)

Player-Level Model with Scoring as Dependant Variable

The Scoring Run variable (p < 0.1) is statistically significant and has a coefficient the size of

0.026. This indicates that momentum accrues as a player keeps scoring in consecutive matches and increases their probability of scoring in the next match. Therefore the longer that a player has scored in consecutive matches, the more momentum he has and the more likely he is to score in the next match.

The conditional Pseudo R-Squared of the model (Table 25) is 27% greater than the marginal Pseudo R-Squared, indicating that the inclusion of the random variables is justified.

The Minutes Played (p < 0.001) variables is statistically significant and has a coefficient of size 0.022. This indicates that the more time a forward is given to play in a match, the higher the likelihood of him scoring in the match (see Figure 35).

The Home variable (p < 0.001) is statistically significant and has a coefficient the size of

0.377. This indicates that forwards are more likely to score at their team’s home ground

(see Figure 35).

Figure 35 Probability of Scoring by Playing Time and Match Venue Player-Level Model with Scoring as Dependant Variable

The Age (p < 0.001) and Age2 (p < 0.001) variables are both statistically significant, with

Age having a coefficient the size of 0.011 and Age Squared having a coefficient the size of

-0.004. This indicates a non-linear relationship between age and the log-odds of scoring; as a forward ages and matures he is more likely to score, this continues until a certain age in which he hits his “goal-scoring peak” around the ages of 27 or 28 and from there his aging seems to decrease his chances of scoring.

The player’s random effects (Figure 38) in the model were in the form of the forward’s random intercept (demonstrating that each forward has a slightly different base-rate for scoring) and random slopes for Scoring Previously (indicating that each player is affected slightly differently by Scoring Previously). The other random effects came from the different seasons in which the match was played (demonstrating that each season had slightly different base-rates of scoring during them).

83 Figure 38 Player’s Random Intercept and Random Slopes of Scoring Previously from optimal Player-Level Model with Scoring as Dependant Variable

3.1.2 Scoring or Assisting a Goal is Considered as Success

Table 28 Measures of Fit: Player-Level Analysis – Scoring or Assisting a Goal is Considered as Success AIC 102498.6 Table 29 Pseudo R-Squared of Player-Level Analysis - Scoring or Assisting a Goal is Considered as Success Pseudo R-Squared Marginal Conditional 0.107 0.134

Table 30 Random Effects of Player-Level Analysis - Scoring a Goal is Considered as Success Groups Variable Name Variance Std. Dev Correlation

84 Players Intercept 0.097 0.311 Scored or Assisted 0.009 0.096 -0.48 Previously Seasons Intercept 0.012 0.112 N = 81,663, groups: Players, 182: Seasons, 56

Table 31 Fixed Effects of Player-Level Analysis – Scoring or Assisting a Goal is Considered as Success Variable Name Estimate Std. Error P Value Intercept -2.895 0.099 < 2e-16 *** Minutes Played 0.022 0.000 < 2e-16 *** Age 0.023 0.004 1.54e-10 *** Age2 -0.004 0.000 < 2e-16 *** Scored or Assisted 0.292 0.099 0.003 ** Previously Age * Scored or -0.009 0.004 0.015 * Assisted Previously Scoring or Assisting Run 0.022 0.011 0.044 * Played Previously -0.011 0.027 0.683 Rest Time -0.002 0.002 0.450 Home 0.410 0.015 < 2e-16 *** Signif codes: 0.001 “ *** ” 0.01 “ ** ” 0.05 “ * ” 0.1 “ ’ ”

The following results were obtained from the optimal model using the model selection method outlined in the previous chapter:

The Scored or Assisted Previously (p < 0.01) variable is statistically significant and a coefficient of 0.292. This suggests that a player scoring or assisting in the previous match does increase their probability of scoring or assisting in the next match. Again, similar to the previous model, this suggests that a momentum effect exists for forwards such that

85 providing a goal in their previous match by either scoring or assisting, increases their likelihood of providing a goal in the following match.

The interaction variable between Age and Scoring or Assisting Previously (p < 0.05) is statistically significant and has a coefficient the size of -0.009. This suggests that Age does play a role as a moderator in the effect size of Scoring or Assisting Previously; as a forward’s age increases and he matures, the momentum effect from scoring or assisting in the previous match is weakened (see Figures 40 and 41).

Figure 40

Probability of Scoring or Assisting Away by Age and Scoring or Assisting Previously (Played 90 Minutes)

Player-Level Model with Scoring or Assisting as Dependant Variable

86 Figure 41

Probability of Scoring or Assisting at Home by Age and Scoring or Assisting Previously (Played 90 Minutes)

Player-Level Model with Scoring or Assisting as Dependant Variable

The Scoring or Assisting Run variable (p < 0.05) is statistically significant and has a coefficient the size of 0.022. This indicates that momentum accrues as a player is involved in his team’s goals in consecutive matches and increases their probability of scoring or assisting in the next match.

The conditional Pseudo R-Squared of the model (Table 28) is 25% greater than the marginal Pseudo R-Squared, indicating that the inclusion of the random variables is justified.

The Home variable (p < 0.001) is statistically significant and has a coefficient the size of

0.410. This indicates that forwards are more likely to score or assist at their team’s home ground (see Figure 39).

87 Figure 39 Probability of Scoring or Assisting by Playing Time and Match Venue Player-Level Model with Scoring or Assisting as Dependant Variable

The Age (p < 0.001) and Age2 (p < 0.001) variables are both statistically significant, with

Age having a coefficient the size of 0.023 and Age Squared having a coefficient the size of

-0.004. This indicates a non-linear relationship between age and the log-odds of scoring or assisting; as a forward ages and matures he is more likely to score, this continues until a certain age in which he hits his “goal-providing peak” and from there his aging seems to decrease his chances of scoring.

The player’s random effects (Figure 42) in the model were in the form of the forward’s random intercept (demonstrating that each forward has a slightly different base-rate for scoring or assisting) and random slopes for Scoring or Assisting Previously (indicating that each player is affected slightly differently by Scoring or Assisting Previously). The other random effects came from the different seasons in which the match was played

(demonstrating that each season had slightly different base-rates of scoring or assisting during them).

88 Figure 42 Player’s Random Intercept and Random Slopes of Scoring or Assisting Previously from optimal Player-Level Model with Scoring or Assisting as Dependant Variable

3.1.3 Mediation of Relationships by Minutes Played

During the analysis, we found a potential mediating variable in the form of Minutes

Played that partially explains the momentum effects found of previous success on future success.

A forward is more likely to be given extended opportunities of play in the team if he has been involved in his team’s goals recently and thus considered to be “in-form”, especially when they are younger and their place in the team has not yet been guaranteed.

For forwards, this will ultimately be based on the number of their team’s goals that that they have been a part of. Therefore, a forward who has been involved in providing their team a goal in his previous appearance is more likely to receive more minutes to play in his team’s following match – and as both of the player-level models showed, more minutes leads to a higher likelihood of scoring and/or assisting. These results were present in both player-level models with their respective definitions of success.

89 The first fixed effects linear regression model (Table 32) shows that for a forward, scoring in the previous match leads to playing more minutes in the following match. It also shows that forwards receive more playing minutes as they grow older, and that the effect of scoring in the previous match on future minutes played gets weaker as the player ages.

Table 32 Linear Regression of Minutes Played dependent on Previous Scoring Success, Age and their Interaction Variable Name Estimate Std. Error P Value Intercept 42.260 0.673 < 2e-16 *** Scored Previously 29.510 1.550 < 2e-16 *** Age 0.224 0.025 < 2e-16 *** 퐴푔푒2 -0.002 38.210 < 2e-16 *** Age * Scored Previously -0.353 0.057 7.64e-10 *** Signif codes: 0.001 “ *** ” 0.01 “ ** ” 0.05 “ * ” 0.1 “ ’ ”

The next logistic regression (Table 33) shows that the probability of scoring increases as the player plays more minutes.

Table 33 Logistic Regression of Scoring dependent on Minutes Played Variable Name Estimate Std. Error P Value Intercept -2.445 0.032 < 2e-16 *** Minutes 0.022 0.000 < 2e-16 *** Signif codes: 0.001 “ *** ” 0.01 “ ** ” 0.05 “ * ” 0.1 “ ’ ”

The final logistic regression model (Table 34) indicates that the probability of scoring in the next match increases as the player matures, and if the player had scored in the previous match. Finally we can again see that the previous success to future success relationship weakens as the forwards mature and their age increases – irrelevant of the number of minutes that the forwards play in the next match.

90 Table 34 Logistic Regression of Scoring dependent on Previous Scoring Success, Age and their Interaction Variable Name Estimate Std. Error P Value Intercept -4.389 0.232 < 2e-16 *** Scored Previously 0.602 0.100 1.90e-09 *** Age 0.26 0.017 < 2e-16 *** 퐴푔푒2 -0.005 0.000 < 2e-16 *** Age * Scored Previously -0.012 0.004 0.002 ** Signif codes: 0.001 “ *** ” 0.01 “ ** ” 0.05 “ * ” 0.1 “ ’ ”

Similar results were found with Scoring or Assisting as the definition for success. The underlying model can be seen below in Figure 43.

Figure 43 Underlying Model of the Mediation of Previous Success on Future Success relationship by Minutes Played

Player’s Minutes Played in Next Age Match

Success in Previous Success in Next Match Match

Player’s Age

91 3.2 Team-Level Analysis

In the team-level analysis we set out to research one hypothesis regarding momentum in football:

➢ Does winning the previous league match increase the probability of winning the following

match?

The research was again carried out two times:

1. Accounting for the participating teams’ “quality gap” in the form of the difference between

their respective Elo Rankings prior to the match

2. Without accounting for the participating teams’ “quality gap”

3.2.1 Winning Equals Success – with Team’s Quality Gap

The optimal model of the team-level analysis, accounting for the participating teams’ quality gap:

Table 35 Measures of Fit: Team-Level Analysis – Accounting for Quality Gap AIC 61865.4

Table 36 Pseudo R-Squared of Team-Level Analysis – Accounting for Quality Gap Pseudo R-Squared Marginal Conditional 0.191 0.192 Table 37 Random Effects of Team-Level Analysis – Accounting for Quality Gap Groups Variable Name Variance Std. Dev Teams Intercept 0.003 0.057 N = 52,817, groups: Teams, 178

92 Table 38 Fixed Effects of Team-Level Analysis – Accounting for Quality Gap Variable Name Estimate Std. Error P Value Intercept -1.139e+00 2.247e-02 < 2e-16 *** Elo Difference 4.345e-03 8.426e-05 < 2e-16 *** Home 9.205e-01 2.135e-02 < 2e-16 *** Match History 3.191e-02 1.214e-02 0.009 ** Won Previous League 3.160e-02 2.104e-02 0.133 Match Signif codes: 0.001 “ *** ” 0.01 “ ** ” 0.05 “ * ” 0.1 “ ’ ”

The conditional Pseudo R-Squared (Table 36) is only slightly greater than the marginal

Pseudo R-Squared, suggesting that the inclusion of the random variables is perhaps not justified.

The Won Previous League Match variable was included in the final model based on the

AIC model selection process. Yet, based on conventional methods of determining statistical significance, this variable is not statistically significant (p>0.1). and we therefore based on this model we cannot state that a momentum effect exists on a team-level in football.

The Elo Difference variable (p<0.001) is statistically significant and has a coefficient size of

0.004. This indicates that the as the quality gap of the team grows relative to its competitor, so does its probability of winning the match.

The Home variable (p<0.001) is statistically significant with a coefficient size of 0.921. This suggests that playing at home does give a team a certain advantage in terms of their likelihood to win the match.

The Match History variable (p<0.01) is statistically significant, having a coefficient the size of 0.032. This indicates that previous results between two teams at a specific venue do

93 seem to affect the probability of winning the future between the teams at the same venue.

The random effects in the model (Figure 44) were in the form of the teams’ random intercept (each team has a different base-rate for winning). Random effects for different seasons were omitted from the model as they did not improve the model in terms of its

AIC criteria.

Figure 44 Team’s Random Intercept from optimal team-Level Model – Accounting for Quality Gap

3.2.2 Winning Equals Success – without Team’s Quality Gap

The optimal model of the team-level analysis, without accounting for the participating teams’ quality gap:

Table 39 Measures of Fit: Team-Level Analysis – Without Accounting for Quality Gap AIC 64497.0

Table 40

94 Pseudo R-Squared of Team-Level Analysis – Without Accounting for Quality Gap Pseudo R-Squared Marginal Conditional 0.070 0.114 Table 41 Random Effects of Team-Level Analysis – Without Accounting for Quality Gap Groups Variable Name Variance Std. Dev Teams Intercept 0.164 0.405 N = 52,817, groups: Teams, 178

Table 42 Fixed Effects of Team-Level Analysis – Without Accounting for Quality Gap Variable Name Estimate Std. Error P Value Intercept -1.622 0.052 < 2e-16 *** Home 0.772 0.021 < 2e-16 *** Match History 0.218 0.011 < 2e-16 *** Europe Before 0.092 0.033 0.005 ** Europe After 0.119 0.034 0.000 *** Weighted Asverage 0.101 0.02 5.17e-07 *** League Points League Rest Time -0.000 0.000 0.276 Won Previous League 0.088 0.033 0.007 ** Match Won Previous Match 0.008 0.0323 0.816 Signif codes: 0.001 “ *** ” 0.01 “ ** ” 0.05 “ * ” 0.1 “ ’ ”

The Weighted Average League Form variable (p<0.001) is statistically significant, having a coefficient the size of 0.101. This indicates that recent form over the last four league matches does affect the probability of winning the next league match, and that a weak

95 momentum effect exists on a team-level in football that depends on the team’s results in their previous four league matches.

The Won Previous League Match (p<0.01) variable is also statistically significant, having a coefficient the size of 0.088. This indicates that winning the previous match in the league increases the probability of winning the next league match, and that a weak momentum effect exists on a team-level in football.

The League Rest Time variable and the Won Previous Match variable were both not statistically significant, indicating that a match’s result from outside of the league, does not have a momentum effect in terms of increasing the likelihood of winning the team’s next league match.

The conditional Pseudo R-Squared (Table 40) is greater than the marginal Pseudo R-

Squared, suggesting that the inclusion of the random variables is justified.

The Home variable (p<0.001) is statistically significant with a coefficient size of 0.772, slightly lower than in the previous model. This again suggests that playing at home does give a team a certain advantage in terms of their likelihood to win the league match.

The Match History variable (p<0.001) is statistically significant, having a coefficient the size of 0.218, much greater than in the previous model. This indicates that Match History replaces the Elo Difference and provides some information about the quality gap of 2 opponents in a specific venue. Therefore, if a team has a favourable record against a specific team home or away, this record is likely to be kept in the non-distant future i.e. teams’ quality gap is a slowly changing factor.

The Europe Before variable (p<0.01) and Europe After variable (p<0.001) are both statistically significant, having a coefficient the size of 0.092 and 0.119, respectively. This indicates that playing in Europe before or after is a match leads to a higher probability of

96 winning the league match. The positive coefficient size may be explained by this variable also providing some information about the quality gap between teams, as only the best

(and therefore most high-quality teams) from the previous season will have an opportunity to play in European competitions in the next season.

The random effects in the model (see Figure 45) were in the form of the teams’ random intercept (each team has a different base-rate for winning). The Random effects for the different seasons were omitted from the model as they did not improve it in terms of AIC.

Figure 45 Team’s Random Intercept from optimal team-Level Model – Without Accounting for Quality Gap

3.2.3 Comparison of the Team-Level Models

The two team-level models are comparable because they are both modelling the same dependent variable – winning. Based on the AIC criterion, the model which accounts for the Quality Gap is smaller. Therefore, we can say that the better model of the two alternatives is the model which accounts for the teams’ quality gap.

97 Because a momentum effect was not found in the better of the two models, we cannot decisively determine whether a momentum effect exists on a team-level or not.

This also suggests that the other variables in the model which does not account for the quality gap do capture some of the variance that is provided by the Elo Rating Difference variable but not as much.

98 Chapter 4

Discussion

This study was motivated to put a spotlight on a prevalent belief of momentum in sports.

Surprisingly, research of the momentum phenomenon was largely non-existent in football, and therefore this research is the first to be carried out in a quantitative methodology. Perhaps this was due to football being a uniquely low-scoring sport which warranted a reconsideration of momentum, exploring its prevalence between-matches in opposed to within-matches as has largely been the case in other sporting fields.

Our aim was to discover if the momentum phenomenon has existed empirically in football – carrying out an explorational, observational research based on historical data.

We decided to split our analysis into two relevant levels, player-level and team-level in order to research whether previous success affects future success at both levels.

At the team level we considered success to be winning a football match, while on the player level we decided to look only at players who play in a position with a well-defined and easily measurable definition of success – forwards and their goal contributions to the team.

With this in mind, we set out to collect a rich amount of historical data that spanned over many seasons so that more advanced statistical techniques could be undertaken to answer our research questions – without attempting to explain the underlying psychological explanations and causes that drive this momentum.

In both analysis levels, there was one common factor that was shown to be significant – home advantage. Our research at both levels showed that probability of success was greater when a team or player is playing in their home stadium in front of a majority of

99 their own supporters, and with the comfort of familiar surroundings and with a minimal commute.

4.1 Player-Level

At the player-level, we sought to explore whether forwards are affected by momentum that they gain from carrying out their duties and providing goals for their teams.

Specifically, we asked if forwards gain momentum from scoring goals in a match which they then carry on to their next match and leads to them having a higher likelihood of providing goals in said match. We also wanted to discover if such an effect was stable throughout a forward’s career, or whether it varies as the forwards gain experience over the course of their careers and become experts in providing goals.

Our analyses at the player level have found evidence for the prevalence of a momentum effect in elite football forwards – whether we only considered scoring a goal to be defined as a success but also when we added goal involvement in the form of assists to also be defined as success. Our results also point to a potential build-up of momentum in the cases where a forward is providing goals in successive matches, which also increases their likelihood of providing goals in their next match. This falls in line with the duration effects of Iso-Ahola and Dotson’s (2014) success breeds success models and could be an interesting direction to research in the future.

We have found that momentum from goal involvement is affected by a player’s experience level in two different ways.

Firstly, the likelihood of being involved in a goal is not linear during a forward’s career – it increases gradually during the first half or so of their career as they gain experience and

100 reach their physical peak, but then steadily decreases as the forward matures towards his retirement age.

Secondly, the momentum harnessed from being involved in a goal is strongest at the beginning of a forward’s career and decreases steadily until it becomes non-existent towards the end of a forward’s playing days when he is an expert goal provider. The second effect promotes Bandura (1994) theory that as expertise is gained, self-efficacy stabilizes towards the middle years of life and people settle into established routines in their major areas of functioning.

The most unsurprising insight from the player-level analysis was that there is a positive relationship between playing time and goal involvement. It is only intuitive that the longer that a forward is on the pitch, the more likely he is to have goal providing opportunities and to convert one or more into a goal.

This brings up the possible necessity of teaching emotional regulation at an early age, so that young forwards do not become dependent on scoring in order to believe in themselves and be confident in upcoming matches. Likewise the emotional regulation should also deal with the other end of the spectrum, of bringing them “down-to-earth” and not too self-confident after scoring so that when the scoring inevitably decreases for periods of time, they are not heavily impacted. This will lead to the development of a mentally-stronger forward, who is not dependent on previous performances and has a deep-lying, and balanced belief in his own abilities. Additional research should be carried out in order to pursue this idea further.

4.1.1 Mediation by Minutes Played

As a by-product of this study, we found evidence that the momentum effects found are mediated by the game time that a forward is given after a successful appearance. We

101 found that forwards are likely to be given more match time in an upcoming match if they had been involved in a goal in their previous appearance – and more playing time then leads to a greater likelihood of scoring in that upcoming match.

This should be further researched in the future. Perhaps previous success leads to more playing time which leads to greater likelihood of success, perhaps this rewarding of more playing time after being involved in a goal or punishment of less playing time had they not early in a forward’s career creates this momentum phenomenon by operant conditioning or perhaps it is our society, the coaches, and managers who put so much emphasis on goals that they influence players at a young age and prime them for the momentum phenomenon.

4.2 Team-Level

At the Team-Level, we found minor evidence for the prevalence of momentum. Due to the fact that this research was purely observational, and because the momentum effects that were found were weak, we cannot determine whether momentum exists on a collective team-level in elite European football leagues.

As part of the team-level analysis, we had constructed two models, one containing the Elo

Difference as a quantification of the quality gap between teams and the second purely based on observable information prior to a match’s kick-off.

It is in the latter of the two models, where we found some evidence for a momentum phenomenon. A team that had won their previous league match was found to have a larger probability of a future win in their upcoming match, as did the form that a team had from their previous four matches. Although these effects were weak, they still point to a minor advantage that a team carries into a match if its recent results had been successful.

102 On the other hand, the first model that did incorporate a measure of the quality gap between teams was the better of the two models – both in terms of simplicity and in terms of goodness of fit. In this model though, previous successes were not statistically significant in explaining future successes.

Yet, when we consider how we measured the teams’ quality and their quality gap – specifically how the two participating teams’ Elo Ratings are calculated and updated match by match as part of the Elo Algorithm – it could possibly be considered that an Elo

Rating is just a complex method for remembering previous results and updating a team’s ranking accordingly. Winning a match leads to incremental changes in each of the participating team’s Elo Ratings, such that the winning team’s rating increases the same amount that the losing team’s rating decreases, and this amount is a function of their difference in their Elo Ratings, pre-match.

For example, if a random team was to be on a 4 match winning streak against stronger opposition, then their current Elo Rating compared to their rating from before the streak would be considerably greater. Consider that without being on the streak, the team would have hypothetically had an Elo Rating of 1,750 compared to a more respectable Elo

Rating of 1,850 if they had been on the streak. In their next match, the team was playing against a 1,800 Elo Rated team. The Elo Difference heading into the match would be +50 if they had been on a 4 match winning streak, and -50 had they not been on the winning streak. Our model would then predict a higher probability of victory for the team if they had been on the winning streak. This could possibly mean that Elo Ratings are simply an aggregation over time of different results and streaks, which allow us in the end to quantify a team’s level of quality.

The other factors that were found to be statistically significant in the second model included participation in European competitions, which again could be considered a

103 variable that measures a team’s quality as only the best teams from the previous season can compete in Europe. Thus, playing in Europe has a positive effect on the probability of winning, in contrast to what we theorised that it could lead to fatigue within the team.

The teams’ historical results, when playing their specific opponents in a specific location, was also found to influence the probability of winning. This again could be considered to measure a slowly changing quality gap between the competing teams. This is logical, as if team A was better than team B two years ago it is highly likely that quality gap between the teams has not varied greatly since. Even the teams which received a high influx of investment such as Chelsea and Manchester City did not transform their results overnight.

4.3 Future Research

Momentum in general, and in football specifically, is still a riddle. At both analysis levels, further research is required to understand the momentum phenomenon and its underlying causes.

Qualitative or experimental research should be undertaken in the field of football for a more comprehensive answer to be obtained. By using surveys or experiments and understanding what psychological sensations occur and the root causes that drive the momentum that we observed in our results, we could better advance the field of sports psychology in the sport.

Having been motivated to carry out this research based on press conferences in which goal droughts were discussed, this study did not delve into the effects of negative momentum – that is how non-success affects success in the future. Does negative momentum accumulate and hamper forwards’ confidence, becoming difficult to break out of like Sir Alex Ferguson and Jose Mourinho stated and thus creating goal-droughts.

104 Our study was based solely on between-match momentum, that is how success in one match affects the likelihood of success in the future. Historically though, momentum in other sporting fields has been studied within-matches. Future research should be carried out in football, exploring momentum within-matches; looking at how successful actions vary according to previous pass completion, shooting accuracy, pressing events, etc. More advanced footballing metrics that have been adopted such as Expected Goals (xG),

Expected Assists (xA), and player ratings could also be used to research within-match momentum.

Having seen the effects that age and experience have on momentum in a player-level, it is only logical that the next step in future research would be to incorporate football team’s age distribution into the team-level analysis. Perhaps older teams are less volatile in their reactions to good or bad results, while younger teams are less consistent and more volatile as they are affected by their recent results.

There are limitations in our research that arise from the data that we collected For the team-level, we only studied teams from the “Top 5” European football leagues. If we want to state that the momentum phenomenon is prevalent for football teams in general, then other leagues also need to be analysed. In the player-level analysis, we only looked at the very elite football forwards. Again, if we wish to declare that such momentum is prevalent all football forwards, then further research which incorporates non-elite forwards is required – perhaps this momentum only exists for elite forwards and is one of the factors that separates them from the rest.

The Elo Differences were not included as part of the player-level analysis, mainly because

Elo Ratings for the years in which some of the legends played was not easily attainable.

But it is logical to consider that much like the quality gap between two teams influences the probability of a team to win, it could have a similar effect on the probability of the

105 team’s forward to score in that match. Further research should be carried out, that includes the Elo Difference metric into future the player-level studies.

4.4 Practical Applications

4.4.1 Comparison of Forwards Across Different Eras

Using our models, we can predict a forward’s probability to score in a match. If we aggregate these probabilities, we can get an average probability to score per player, irrelevant of when and where they played. This might not be the most exact method to rank forwards but is nonetheless interesting to explore. The top 10 forwards in football based on their average predicted probabilities to score throughout their careers according to our models can be seen in Table 43. At the top of the list, we can find a familiar name in Lionel Messi. Unfortunately, Diego Maradona’s raw data was not retrieved as it was largely incomplete, as was Pele’s – so unfortunately we could not include them in the comparison.

Table 43 Top 10 Forwards According to Average Predicted Probably to Score During Careers Player Average Probability to Score Lionel Messi 55% Gerd Müller 53.4% Ronaldo 48.7% Cristiano Ronaldo 48.2% Edinson Cavani 46.8% Ruud van Nistelrooy 46.7% Marco van Basten 46.3% Luis Suarez 46.2% Robert Lewandowski 45.8% Zlatan Ibrahimović 45.2%

106 4.4.2 Scouting

Clubs could incorporate this insight of how previous success can affect future success when scouting potential recruits. If a club is looking at a young player who is inconsistent, they can delve deeper into his record and see if he has periods of good form or bad form and account for this. An inconsistent young player may simply need a manager or coach who instils confidence in them and helps educate them to better regulate their emotions, leading said player to be more consistent and causing his performances to be less affected by previous ones. Opposition scouting could also incorporate the insights of this study, placing greater emphasis on opposition players who have had good recent performances.

4.4.3 Manager’s Press Conferences

Football managers are obliged to attend press conferences pre and post-match, answering questions laid out to them by sports journalists. Questions about the team’s form and the player’s form is almost inevitable. This provides a special opportunity though for the managers – to publicly back their players and make their belief in a player’s ability evident for everyone to see, including the player in question who inevitably is aware of what is written about him in the press or the social networks. The effects of such public support can be very powerful for some players, depending on their character of course.

4.4.4 Man Management

Managers should take the mediation effect of playing time that we discovered during the player-level analysis into account, especially when dealing with younger inexperienced players. These younger players should be encouraged, motivated, and instilled with confidence during patches of bad form and not dropped from the team as a response. On

107 the other hand, during good patches of form, these young players should be rested somewhat so that they don’t become primed to believe that when they play well they will automatically be given more playing time. It is during these good patches of form that these young players should be “brought down to earth” and taught to self-regulate their emotions, whether they are positive or negative.

4.4.5 Story-Telling of Sports

Putting a compelling narrative on any story makes it more interesting and worth sharing, increasing the interest in the sport, a specific match, or a player. There is no reason for sports commentators not to include a momentum effect in their sports journalism. Firstly, it is difficult for the consumers of their product (the sports fans) to debunk or confirm the effects of the momentum which they write about, many academic studies have tried.

Secondly, this momentum effect is embedded in many a sports fans’ perspective of sports

– form is a mainstay of sporting discussions. Finally, momentum and form can be an explanation for deeper-lying factors that were not incorporated in this study such as injuries and suspensions to key players, or unrest within a team’s dressing room. These factors are usually known to the journalists who combine these happenings together with form to write captivating sports articles.

108 Appendices

Appendix A The 95th Percentile of Goal Scorers in the “Top-5” European Football Leagues

Cristiano Ronaldo Anthony Modeste Steven Gerrard Kylian Mbappé Lionel Messi Claudio Pizarro Robinho Emiliano Sala Edinson Cavani Salomon Kalou Dirk Kuyt Jimmy Briand Sergio Agüero Marco Reus Chicharito Ivan Santini Benjamin Zlatan Ibrahimovic Mario Gómez Clint Dempsey Moukandjo Martin Gonzalo Higuaín Bafétimbi Gomis Papiss Demba Cissé Braithwaite André-Pierre Edin Dzeko Gignac Michu Ireneusz Jelen Antonio Di Natale Lisandro López Rickie Lambert Mamadou Niang Wissam Ben Robert Lewandowski Yedder André Ayew Youssouf Hadji Pierre-Emerick Aubameyang Mohamed Salah Gerard Moreno Gervinho Wayne Rooney Jamie Vardy Diego Forlán Dimitri Payet Luis Suárez Heung-min Son Giuseppe Rossi Neymar Olivier Giroud Darren Bent Pedro Mauro Icardi Fernando Llorente Frank Lampard Carlos Vela Fabio Quagliarella Karim Benzema Jermain Defoe Imanol Agirretxe Francesco Totti Kévin Gameiro Dimitar Berbatov Nolito Giampaolo Pazzini Thomas Müller Demba Ba Youssef El Arabi Roberto Firmino Romelu Lukaku Christian Benteke Dries Mertens Riyad Mahrez Alexandre Lacazette Iago Aspas Andrea Belotti Fernando Torres Emmanuel Eden Hazard Samuel Eto'o Lorenzo Insigne Adebayor Mario Balotelli Raúl Nikola Kalinic Florent Malouda Gareth Bale Ciro Immobile José Callejón Kevin Kuranyi Alexis Sánchez Marco Borriello Marek Hamsik Lukas Podolski Loïc Rémy Iago Falque Giovanni Simeone Florian Thauvin

109 Massimo Antoine Griezmann Paulo Dybala Maccarone Nabil Fekir Carlos Bacca Luca Toni Éder Alassane Plea Roberto Soldado Miroslav Klose Domenico Berardi Aritz Aduriz Diego Milito Adrian Mutu Alberto Gilardino Marco Di Vaio Antonio Cassano Arjen Robben Alexandre Pato Sergio Floccari Vedad Ibisevic Fabrizio Miccoli Alessandro Matri Stefan Kießling Rodrigo Palacio Stevan Jovetic Harry Kane Germán Denis Mattia Destro Carlos Tévez Raffael Sandro Wagner Klaas-Jan Robin van Persie Huntelaar Shinji Okazaki Diego Costa Alexander Meier Timo Werner David Villa Lucas Barrios Andrej Kramaric Álvaro Negredo Mario Mandzukic Max Kruse Falcao Mevlüt Erdinc Patrick Helmes Rubén Castro Nenê Milivoje Novakovic

110 Appendix B Historical Forward Legends of Football Alan Shearer Eric Cantona Jürgen Klinsmann Roberto Baggio Alessandro Del Piero Filippo Inzaghi Kaká Ronaldinho Andriy Shevchenko Gabriel Batistuta Marco van Basten Ronaldo Bernd Schuster George Weah Michael Laudrup Rudi Völler Ruud van Christian Vieri Gerd Müller Michael Owen Nistelrooy David Trézéguet Hernán Crespo Nicolas Anelka Zinédine Zidane Davor Suker Iván Zamorano Paolo Rossi Thierry Henry Dennis Bergkamp Jean-Pierre Papin Patrick Kluivert Wesley Sneijder Diego Maradona Jorge Valdano Rivaldo Youri Djorkaeff

111 Appendix C Teams that played in “Top-5” European football leagues from 2000-2019 1. FC Köln Cesena FC HSC Montpellier SC Freiburg 1.FC Charlton Athletic Huddersfield Town SC Paderborn 07 Kaiserslautern 1.FC Nuremberg Chelsea FC Hull City SCO Angers 1.FSV Mainz 05 Chievo Verona Inter Milan SD Eibar AC Ajaccio Como 1907 Ipswich Town SD Huesca AC Le Havre Córdoba CF Istres Football Club Sevilla FC AC Milan Coventry City Juventus FC Sheffield United AC Perugia Calcio Crystal Palace Karlsruher SC SM Caen CS Sedan- ACD Treviso Leeds United Southampton FC Ardennes Delfino Pescara ACF Fiorentina Leicester City SPAL 1936 ACR Messina Deportivo Alavés Levante UD Sporting Gijón Deportivo de La AFC Bournemouth Liverpool FC SpVgg Greuther Fürth Coruña AJ Auxerre Derby County LOSC Lille SpVgg Unterhaching Albacete EA Guingamp LR Vicenza Virtus SS Lazio Balompié Alemannia Eintracht Málaga CF SSC Bari Aachen Braunschweig Eintracht Arminia Bielefeld Manchester City SSC Napoli Frankfurt Arsenal FC Elche CF Manchester United Stade Brest 29 AS Livorno ES Troyes AC Middlesbrough FC Stade Reims AS Monaco Everton FC Modena FC 2018 Stade Rennais FC AS Nancy-Lorraine FC Augsburg MSV Duisburg Stoke City AS Roma FC Barcelona Newcastle United Sunderland AFC AS Saint-Étienne FC Crotone Nîmes Olympique SV Darmstadt 98 Ascoli Calcio FC Empoli Norwich City SV Werder Bremen FC Energie Aston Villa Novara Calcio 1908 Swansea City Cottbus

112 FC Évian Thonon Atalanta BC OGC Nice Torino FC Gaillard FC Girondins Athletic Bilbao Olympique Lyon Tottenham Hotspur Bordeaux Athlétic Club FC Hansa Rostock Olympique Marseille TSG 1899 Hoffenheim Arlésien Atlético Madrid FC Ingolstadt 04 Paris Saint-Germain TSV 1860 Munich Bayer 04 FC Le Mans Parma Calcio 1913 UC Sampdoria Leverkusen Bayern Munich FC Lorient Piacenza Calcio 1919 UD Almería Benevento Calcio FC Metz Portsmouth FC UD Las Palmas Queens Park Birmingham City FC Nantes Udinese Calcio Rangers Blackburn Rovers FC Schalke 04 Racing Santander Urbs Reggina 1914 FC Sochaux- Blackpool FC Rayo Vallecano US Anconitana Montbéliard Bologna FC 1909 FC St. Pauli RB Leipzig US Boulogne Bolton Wanderers FC Toulouse RC Lens US Lecce Borussia RC Strasbourg FCO Dijon US Palermo Dortmund Alsace Borussia Fortuna RCD Espanyol US Sassuolo Mönchengladbach Düsseldorf Barcelona Bradford City Frosinone Calcio RCD Mallorca Valencia CF Brescia Calcio Fulham FC Reading FC Valenciennes FC Brighton & Hove Genoa CFC Real Betis Balompié Venezia FC Albion Burnley FC Getafe CF Real Madrid VfB Stuttgart CA Osasuna GFC Ajaccio Real Murcia CF VfL Bochum Gimnàstic de Cádiz CF Real Oviedo VfL Wolfsburg Tarragona Cagliari Calcio Girona FC Real Sociedad Villarreal CF Calcio Catania Granada CF Real Valladolid CF Watford FC West Bromwich Cardiff City Grenoble Foot 38 Real Zaragoza Albion Carpi FC 1909 Hamburger SV Recreativo Huelva West Ham United CD Leganés Hannover 96 Robur Siena Wigan Athletic

113 Wolverhampton CD Numancia Hellas Verona SC Amiens Wanderers CD Tenerife Hércules CF SC Bastia Xerez CD Celta de Vigo Hertha BSC

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תק ציר:

הרבה מחקרים עסקו בנושא של מומנטום בספורט, וגררו ויכוחים רבים לגבי תוצאותן השונות. אנו מגדירים

מומנטום להיות מצב שבו הצלחה קודמת באירוע מסויים מובילה להגדלת הסיכויים להצלחה באירוע הבא

הישיר. הויכוח המקורי עסק בעצם התוקף של שימוש במילה מומנטון להגדרת המצב, עם הסקת השוא של "יד-

חמה" בכדורסל (Gilovich, Vallone, & Tversky, 1985) שטענו שהמומנטם שנצפה ונתפס על ידינו הוא רק

היוריסטיקה אנושית שמאפשרת לנו לראות תבניות, אפילו אם התבניות הללו נגרמות מרנדומליות. באופן

מפתיע, התופעה של מומנטום לא נחקרה כמעט בספורט הכי פופולרי בעולם – כדורגל. המחקר הזה הוא מחקר

תצפיתי של נתוני סדרת-זמן של קבוצות כדורגל ושחקני כדורגל מליגות הכדורגל הבחירות של אירופה. מצאנו

ראיות לקיומו של אפקט המומנטום בין משחק למשחק עבור חלוצים כתוצאה מכך שהם סיפקו שער עבור

קבוצתם בהופעתם הקודמת. ברמת הקבוצות, מצאנו אפקט מומנטום חלש שניתן ליחס לאיכות היחסית של

הקבוצה. אנו דנים בהשלכות של הממצאים הללו, ומחקרים עתידיים שצריך לקיים ע"מ לקדם את מחקר

המומנטום בכדורגל.

מילות מפתח: מומנטום, ספורט, כדורגל, יד חמה, רגרסיה לוגיסטית עם אפקטים מעורבים, סטטיסטיקה

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